Numerical Analysis of the Ultrarelativistic and Magnetized Bondi–Hoyle Problem by Andrew Jason Penner B.Sc., University of Manitoba, 2002 M.Sc., University of Manitoba, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2011 c Andrew Jason Penner 2011 ABSTRACT In this thesis, we present numerical studies of models for the accretion of fluids and magnetofluids onto rotating black holes. Specifically, we study three main scenarios, two of which treat accretion of an unmagnetized perfect fluid characterized by an internal energy sufficiently large that the rest-mass energy of the fluid can be ignored. We call this the ultrarelativistic limit, and use it to investigate accretion flows which are either axisymmetric or restricted to a thin disk. For the third scenario, we adopt the equations of ideal magnetohydrodyamics and consider axisymmetric solutions. In all cases, the black hole is assumed to be moving with fixed velocity through a fluid which has constant pressure and density at large distances. Because all of the simulated flows are highly nonlinear and supersonic, we use modern computational techniques capable of accurately dealing with extreme solution features such as shocks. In the axisymmetric ultrarelativistic case, we show that the accretion is described by steadystate solutions characterized by well-defined accretion rates which we compute, and are in reasonable agreement with previously reported results by Font and collaborators [1, 2, 3]. However, in contrast to this earlier work with moderate energy densities, where the computed solutions always had tail shocks, we find parameter settings for which the time-independent solutions contain bow shocks. For the ultrarelativistic thin-disk models, we find steady-state configurations with specific accretion rates and observe that the flows simultaneously develop both a tail shock and a bow shock. For the case of axisymmetric accretion using a magnetohydrodynamic perfect fluid, we align the magnetic field with the axis of symmetry. Preliminary results suggest that the resulting flows remain time-dependent at late times, although we cannot conclusively rule out the existence of steady-state solutions. Moreover, the flow morphology is different in the magnetic case: additional features are apparent that include an evacuated region near the symmetry axis and close to the black hole. ii TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Project Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Numerical Relativistic Hydrodynamics: A Brief Review . . . . . . . . . . . . . . . . 4 1.2.1 Ideal Hydrodynamic Approximation . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Ultrarelativistic Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 1.4 Numerical Relativistic Magnetohydrodynamics: A Review . . . . . . . . . . . . . . 9 1.3.1 Ideal Magnetohydrodynamic Approximation . . . . . . . . . . . . . . . . . . 9 1.3.2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Bondi–Hoyle Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Non-relativistic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.2 Relativistic Regime 1.4.3 Ultrarelativistic Fluid Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Notation, Conventions and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 iii TABLE OF CONTENTS 2 Formalism and Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 3+1 Decomposition 2.3 Black Hole Spacetimes 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 Minkowski or Special Relativistic Spacetime . . . . . . . . . . . . . . . . . . 29 2.3.2 Spherically Symmetric Spacetime 2.3.3 Axisymmetric Spacetime 2.3.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 . . . . . . . . . . . . . . . . . . . . . . . . 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.1 Hydrodynamics, A Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.2 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.3 Relativistic Force Free Condition . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Derivation of The Equations of Motion 2.6 Conservation of the Divergence Free Magnetic Field . . . . . . . . . . . . . . . . . . 46 2.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . 41 Divergence Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7 Ultrarelativistic Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.8 Geometric Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Finite Volume Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Hyperbolic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Calculating the Primitive Variables 3.4 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.1 MHD Wave Mathematical Description . . . . . . . . . . . . . . . . . . . . . 60 3.4.2 MHD Waves and Characteristic Velocities . . . . . . . . . . . . . . . . . . . 63 3.5 Conservative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6.1 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.6.2 Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6.3 Contact Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.7 The Riemann Problem: Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.8 The Godunov Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 iv TABLE OF CONTENTS 3.9 3.8.1 The Relativistic Godunov Scheme . . . . . . . . . . . . . . . . . . . . . . . . 80 3.8.2 Variable Reconstruction at Cell Boundaries . . . . . . . . . . . . . . . . . . . 83 3.8.3 Flux Approximations 3.8.4 Limitations of Approximate Riemann Solvers . . . . . . . . . . . . . . . . . . 92 3.8.5 Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.8.6 The Courant–Friedrichs–Lewy (CFL) Condition 3.8.7 Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 . . . . . . . . . . . . . . . . 94 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.9.1 The Floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4 Numerical Analysis and Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.1.2 Convergence Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2 Independent Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3 Shock and Symmetry Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3.1 Sod Shock Tube Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3.2 Balsara Blast Wave 4.3.3 Magnetized Strong Blast Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.4 Two Dimensional Riemann Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 . . . . . . . . . . . . . . . . . . . . . . . . 113 4.4 Kelvin Helmholtz Instability 4.5 Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.6 Steady State Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.6.1 Spherical Relativistic Bondi Accretion . . . . . . . . . . . . . . . . . . . . . . 122 4.6.2 Magnetized Spherical Accretion 5 Results 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 . . . . . . . . . . . . . . . . . . . . . . . . . 129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Accretion Phenomenon and Accretion Rates . . . . . . . . . . . . . . . . . . . . . . 136 5.1.1 Rest Mass Accretion Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1.2 Stress-Energy Accretion Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.1.3 Energy Accretion Rate 5.1.4 Azimuthal Angular Momentum Accretion Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 . . . . . . . . . . . . . . . . 138 v TABLE OF CONTENTS 5.1.5 5.2 5.3 5.4 Radial Momentum Accretion Rate . . . . . . . . . . . . . . . . . . . . . . . . 139 Axisymmetric Bondi–Hoyle UHD Accretion Onto a Black Hole . . . . . . . . . . . . 140 5.2.1 Axisymmetric Accretion: a=0 . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2.2 Axisymmetric Accretion: a=0 . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Non-axisymmetric Infinitely Thin-Disk UHD Accretion Onto a Black Hole . . . . . 158 5.3.1 Infinitely Thin-Disk Accretion: a=0 . . . . . . . . . . . . . . . . . . . . . . . 160 5.3.2 Infinitely Thin-Disk Accretion: a=0 . . . . . . . . . . . . . . . . . . . . . . . 172 Magnetohydrodynamic Bondi–Hoyle Accretion Onto a Black Hole . . . . . . . . . . 173 5.4.1 Magnetized Axisymmetric Accretion: a=0 . . . . . . . . . . . . . . . . . . . 186 5.4.2 Magnetized Axisymmetric Accretion: a = 0 . . . . . . . . . . . . . . . . . . . 187 6 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.1.1 Ultrarelativistic Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.1.2 Magnetohydrodynamic Accretion . . . . . . . . . . . . . . . . . . . . . . . . 212 6.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Appendix A Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 A.1 Axisymmetric Ultrarelativistic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 A.2 Non-axisymmetric Ultrarelativistic Flow . . . . . . . . . . . . . . . . . . . . . . . . 238 A.3 Axisymmetric Magnetohydrodynamic Flow . . . . . . . . . . . . . . . . . . . . . . . 246 Appendix B Code Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 B.1 Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 B.2 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 B.3 Main Routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 B.3.1 Initialize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 B.3.2 Makestep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 B.3.3 Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 B.3.4 Update Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 vi TABLE OF CONTENTS B.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 vii LIST OF TABLES 4.1 1D Minkowski Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2 2D Minkowski Riemann Shock Tube Test . . . . . . . . . . . . . . . . . . . . . . . . 115 4.3 Kelvin–Helmholtz Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.4 Rigid Rotor Test Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.1 Axisymmetric Ultrarelativistic Accretion Parameters . . . . . . . . . . . . . . . . . . 143 5.2 Axisymmetric Ultrarelativistic Accretion Parameters . . . . . . . . . . . . . . . . . . 148 5.3 Non-axisymmetric Ultrarelativistic Accretion Parameters . . . . . . . . . . . . . . . 158 5.4 Magnetized Spherical Accretion Parameters . . . . . . . . . . . . . . . . . . . . . . . 186 5.5 Magnetized Accretion Parameters, a = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 187 viii LIST OF FIGURES 1.1 Hoyle–Lyttleton Accretion Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Bondi–Hoyle Accretion Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 The 3+1 Decomposition of Relativistic Spacetime . . . . . . . . . . . . . . . . . . . . 27 2.2 A Schematic for the Axisymmetric Spacetime . . . . . . . . . . . . . . . . . . . . . . 32 3.1 A Graphical Representation of a Finite Volume . . . . . . . . . . . . . . . . . . . . . 67 3.2 A Scalar Example of an Initial Data Set for the Riemann Problem . . . . . . . . . . 69 3.3 General Magnetohydrodynamic Characteristic Fan. . . . . . . . . . . . . . . . . . . . 70 3.4 Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5 Rarefaction Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6 A Modified Initial Data For a Rarefied Initial Data . . . . . . . . . . . . . . . . . . . 74 3.7 A Discretization of the Continuum Space . . . . . . . . . . . . . . . . . . . . . . . . 78 3.8 The 2D Cell for the Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . 81 3.9 The Schematic for the Piecewise Linear Schemes . . . . . . . . . . . . . . . . . . . . 85 3.10 The Characteristic Fan for the HLL Flux Approximation . . . . . . . . . . . . . . . 90 3.11 The CFL Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.1 Sod Tube Test at t = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Balsara Blast Wave at t = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3 Shock Waves Experienced in MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.4 Convergence of the Balsara Blast Wave at t = 0.4. . . . . . . . . . . . . . . . . . . . 111 4.5 Convergence of the Balsara Blast Wave at t = 0.4, Magnified. . . . . . . . . . . . . . 112 4.6 One Dimensional Strong Blast Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.7 One Dimensional Strong Blast Wave High Resolution . . . . . . . . . . . . . . . . . . 114 4.8 2D Riemann Shock Tube Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 ix LIST OF FIGURES 4.9 Hydrodynamic Kelvin–Helmholtz Instability . . . . . . . . . . . . . . . . . . . . . . . 118 4.10 Hydrodynamic Kelvin–Helmholtz Instability High Resolution . . . . . . . . . . . . . 119 4.11 Magnetized Kelvin–Helmholtz Instability B x = 0.5, 5.0 . . . . . . . . . . . . . . . . . 120 4.12 Rigid Rotor ψ and ∇ · B Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.13 Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.14 Rigid Rotor Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.15 Spherical Accretion Convergence Test . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.16 Spherical Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.17 Magnetized Spherical Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.18 Magnetic Spherical Accretion Convergence Test . . . . . . . . . . . . . . . . . . . . . 131 5.1 Axisymmetric Relativistic Bondi–Hoyle Accretion Setup . . . . . . . . . . . . . . . . 141 5.2 Ultrarelativistic Pressure Profile in the Upstream and Downstream Regions, v∞ = 0.7, 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.3 Axisymmetric Ultrarelativistic Accretion rates, Γ = 4/3 . . . . . . . . . . . . . . . . 147 5.4 Ultrarelativistic Pressure Profile Upstream and Downstream, for v∞ = 0.6. . . . . . 148 5.5 Ultrarelativistic Convergence Angular Cross Section . . . . . . . . . . . . . . . . . . 149 5.6 Ultrarelativistic Convergence Radial Slice . . . . . . . . . . . . . . . . . . . . . . . . 150 5.7 Ultrarelativistic Accretion Onto a Spherically Symmetric Black Hole Model U1 . . . 151 5.8 Ultrarelativistic Accretion Onto a Spherically Symmetric Black Hole Model U2 . . . 152 5.9 Ultrarelativistic Accretion Onto a Spherically Symmetric Black Hole Model U4 . . . 153 5.10 Ultrarelativistic Pressure Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.11 Ultrarelativistic Pressure Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.12 Ultrarelativistic Pressure Profile Convergence Test . . . . . . . . . . . . . . . . . . . 155 5.13 Energy Accretion Rates for v∞ = 0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.14 Energy Accretion Rates for v∞ = 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.15 Non-axisymmetric Relativistic Bondi–Hoyle Accretion Setup . . . . . . . . . . . . . . 159 5.16 Ultrarelativistic Energy Accretion Rates, Γ = 4/3 . . . . . . . . . . . . . . . . . . . . 161 5.17 Ultrarelativistic Upstream Pressure Profile, Γ = 4/3 a = 0 . . . . . . . . . . . . . . . 162 5.18 Ultrarelativistic Upstream Pressure Profile, Γ = 4/3, rmax = 1000 . . . . . . . . . . . 163 5.19 Ultrarelativistic Accretion Onto a Spherically Symmetric Black Hole . . . . . . . . . 164 5.20 UHD Infinitely Thin-Disk Accretion Pressure Profile . . . . . . . . . . . . . . . . . . 165 x LIST OF FIGURES 5.21 Ultrarelativistic Energy Accretion Rate, a = 0 . . . . . . . . . . . . . . . . . . . . . . 166 5.22 UHD Accretion Onto a Spherically Symmetric Black Hole v∞ = 0.9 . . . . . . . . . 167 5.23 UHD Accretion Onto a Spherically Symmetric Black Hole v∞ = 0.9 Interior . . . . . 168 5.24 A Comparison Between rmax = 50 and rmax = 1000 Pressure Fields . . . . . . . . . . 169 5.25 Ultrarelativistic Angular Momentum Accretion Rate, a = 0 . . . . . . . . . . . . . . 170 5.26 Ultrarelativistic Angular Momentum Accretion Rate Convergence, a = 0 . . . . . . . 171 5.27 Ultrarelativistic Angular Momentum Accretion Rate, a = 0.5 . . . . . . . . . . . . . 174 5.28 Ultrarelativistic Infinitely Thin-disk Accretion v∞ = 0.9, a = 0.5, rmax = 50 . . . . . 175 5.29 Ultrarelativistic Infinitely Thin-disk Accretion v∞ = 0.6, a = ±0.5, rmax = 1000M . 176 5.30 Ultrarelativistic Infinitely Thin-disk Energy Accretion Rates v∞ = 0.6, a = ±0.5, rmax = 1000M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.31 Ultrarelativistic Infinitely Thin-disk Azimuthal Angular Momentum Accretion Rates v∞ = 0.6, a = ±0.5, rmax = 1000M . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.32 Ultrarelativistic Infinitely Thin-disk Radial Momentum Accretion Rates v∞ = 0.6, a = ±0.5, rmax = 1000M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.33 Ultrarelativistic Infinitely Thin-disk Accretion Rates v∞ = 0.6, 0.9, a = 0, 0.5, rmax = 1000M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.34 Ultrarelativistic Infinitely Thin-disk Accretion Rates v∞ = 0.6, 0.9, a = 0, 0.5, rmax = 1000M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.35 Ultrarelativistic Energy Accretion Rate a = 0.5. . . . . . . . . . . . . . . . . . . . . . 182 5.36 Ultrarelativistic Radial Momentum Accretion Rate a = 0.5. . . . . . . . . . . . . . . 183 5.37 Ultrarelativistic Radial Momentum Accretion Rate a = 0, 0.5. . . . . . . . . . . . . . 184 5.38 Magnetized Axisymmetric Relativistic Bondi–Hoyle Accretion Profile 1 . . . . . . . . 188 5.39 Magnetized Axisymmetric Relativistic Bondi–Hoyle Accretion Profile 2 . . . . . . . . 189 5.40 Magnetized Axisymmetric Relativistic Bondi–Hoyle Accretion Total Pressure Profile 190 5.41 Magnetized Axisymmetric Relativistic Bondi–Hoyle Accretion . . . . . . . . . . . . . 191 5.42 Magnetized Axisymmetric Relativistic Bondi–Hoyle Accretion Profile 2 . . . . . . . . 192 5.43 Magnetized Axisymmetric Relativistic Accretion Total Pressure for Model M2 . . . . 193 5.44 Magnetized Relativistic Accretion Pressure Cross Sections for Model M1 . . . . . . . 194 5.45 Magnetized Axisymmetric Relativistic Accretion Pressure Profiles for Model M1 . . 195 5.46 Convergence Test Axisymmetric Relativistic Magnetic Accretion . . . . . . . . . . . 196 xi LIST OF FIGURES 5.47 Magnetized Relativistic Accretion Pressure Cross Sections for Model M2 . . . . . . . 197 5.48 Magnetized Axisymmetric Relativistic Accretion Pressure Profiles for Model M2 . . 198 5.49 Magnetized Axisymmetric Relativistic Accretion, B∞ = Bz . . . . . . . . . . . . . . 199 5.50 Magnetized Axisymmetric Relativistic Accretion, B∞ = Bz . . . . . . . . . . . . . . 200 5.51 Magnetohydrodynamic Thermal Pressure Cross Section, a = 0.5, v∞ = 0.9. . . . . . 201 5.52 Magnetized Axisymmetric Relativistic Bondi–Hoyle Accretion . . . . . . . . . . . . . 202 5.53 Magnetized Axisymmetric Relativistic Bondi–Hoyle Accretion Profile 3 . . . . . . . . 203 5.54 Magnetized Axisymmetric Relativistic Accretion Total Pressure for Model M3 . . . . 204 5.55 ||Ψ(t, r, θ)||2 Axisymmetric MHD Bondi–Hoyle Accretion . . . . . . . . . . . . . . . . 205 5.56 Convergence Test Axisymmetric Relativistic Magnetic Bondi–Hoyle Accretion . . . . 206 5.57 Pressure Cross Section at r = 2M for Model M3 . . . . . . . . . . . . . . . . . . . . . 207 5.58 Magnetized Axisymmetric Relativistic Accretion, B∞ = Bz . . . . . . . . . . . . . . 208 5.59 Magnetized Axisymmetric Relativistic Accretion, B∞ = Bz . . . . . . . . . . . . . . 209 5.60 Magnetized Axisymmetric Relativistic Accretion, B∞ = Bz , a = 0.5 . . . . . . . . . 210 A.1 The Axisymmetric Accretion Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 A.2 Major Features of the Axisymmetric Flow Profiles. . . . . . . . . . . . . . . . . . . . 231 A.3 Time Evolution for Model U2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 A.4 Time Evolution for Model U4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 A.5 Time Evolution for Model U7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 A.6 Time Evolution for Model U7 Continued . . . . . . . . . . . . . . . . . . . . . . . . . 235 A.7 Time Evolution for Model U11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 A.8 Time Evolution for Model U11 Continued . . . . . . . . . . . . . . . . . . . . . . . . 237 A.9 The Non-axisymmetric Infinitely Thin-Disk Accretion Setup. . . . . . . . . . . . . . 238 A.10 Major Features of the Infinitely Thin-Disk Accretion Profiles. . . . . . . . . . . . . . 239 A.11 Time Evolution of the Thin-disk Accretion for Model U12 . . . . . . . . . . . . . . . 240 A.12 Time Evolution of the Thin-disk Accretion for Model U12 Cont. . . . . . . . . . . . 241 A.13 Time Evolution of the Thin-disk Accretion for Model U13 . . . . . . . . . . . . . . . 242 A.14 Time Evolution of the Thin-disk Accretion for Model U12 With rmax = 1000 . . . . 243 A.15 Time Evolution of the Thin-disk Accretion for Model U13 With rmax = 1000. . . . . 244 A.16 Shock Capturing Properties of the UHD Shock Capturing Code. . . . . . . . . . . . 245 A.17 Time Evolution of the Thin-disk Accretion for Model M1 . . . . . . . . . . . . . . . 247 xii LIST OF FIGURES A.18 Time Evolution of the Thin-disk Accretion for Model M2 . . . . . . . . . . . . . . . 248 A.19 Time Evolution of the Thin-disk Accretion for Model M3. . . . . . . . . . . . . . . . 249 xiii ACKNOWLEDGEMENTS First I would like to thank my supervisor Matt Choptuik, without his guidance and determination, I would not be the student that I am today. His continuous efforts to mould us into computational physicists is what will pave the way for future research in physics. I owe a debt of gratitude to Dr. W. Unruh, Dr. J. Heyl, and Dr. D. Jones for their involvement and support in different aspects of my research project, as well as the various group meetings that expanded my understanding of general relativity and astrophysics. Further to this I would like to thank the astronomers at the University of British Columbia for their input and educational guidance during my stay. I wish to thank everyone who helped me get where I am today, starting with my grandparents, Andy and Lorna Moffat, and parents, Jim and Doreen Penner, all of whom showed me that hard work will pay off in dividends. This list has to include the employers who hired me along the way, showing me so many life skills that I will continue to use. Unfortunately, due to space considerations, I cannot name all those who helped individually. I certainly thank my friends, who were there for me through thick and thin, to give advice on life, research, and sometimes just a place to go and vent. Wherever I go from here know that you will never be forgotten. I owe a special thanks to my friends Anand Thirumalai and Silke Weinfurtner, who were there for me in some of my most troubling times, and some of my best. Although they commited a great effort to help me through the academic side of life, what is most memorable are the times we just hung out and relaxed. I am happy to thank members of the current and extended research group; Benjamin Gutierrez, Silvestre Aguilar-Martinez, Dominic Marchand, and Ramandeep Gill for all the advice and diversions along the way. I would be remiss if I did not also acknowledge the efforts of past group members such as Dr. D. Neilsen, Dr. S. Noble, Dr. M. Snajdr, Dr. B. Mundim, and Roland Stevenson for their xiv ACKNOWLEDGEMENTS advice throughout this research project and my graduate career. I would also like to thank Gerhard Huisken and the Albert Einstein Institute (MPI-AEI, Germany) for their hospitality and support during the time that I spent there. I also thank the people in the School of Mathematics in the University of Southampton, in specifically for their advice in the later stages of my data analysis. Finally I thank the funding agencies NSERC and CIFAR for their financial support. xv DEDICATION I dedicate this thesis to my mother, she was there for the beginning of this degree, and did not make it to the end. Her love, support, and constant care will never be forgotten. She is very much missed. xvi CHAPTER 1 INTRODUCTION One of the more intriguing aspects of physics is the existence of gravitationally compact objects that curve spacetime. These objects include neutron stars and black holes and are thought to be the driving mechanism behind many interesting astrophysical phenomena, including the build-up of mass via rotating structures known as accretion disks. Gravitationally compact objects have a radius, R, close to their Schwarzschild radius, Rs = 2GM/c2 , where G is Newton’s gravitational constant, and c is the speed of light [4]. Many gravitationally compact objects—including many classes of neutron stars—are thought to be surrounded by accretion disks. Realistic modelling of neutron stars involves complicated microphysics, especially in the interactions between their atmospheres and the accreting matter. On the other hand, black holes, while the most extreme class of gravitationally compact objects, have well defined boundaries that can be modelled without an atmosphere. In both cases, however, the compact objects themselves curve spacetime and, as far as the gravitational interaction goes, are thus most appropriately studied using Einstein’s theory of general relativity. In this thesis, we numerically model scenarios involving the gravitationally mediated accretion of matter onto a black hole. Astrophysically, the matter is expected to be a highly ionized fluid, or plasma. In general, direct modelling of the plasma degrees of freedom is prohibitively expensive computationally: a hydrodynamic approximation is thus frequently made, and we will follow this approach here. The effects of magnetic fields are expected to be important in the accretion problems we consider, and we thus include some of these effects via the so called ideal magnetohydrodynamic approximation, wherein the plasma is assumed to have infinite conductivity. We further assume that the spacetime containing the black hole is fixed (non-dynamic), which is tantamount to asserting that the accreting fluid is not self-gravitating.1 Of the many types of black-hole-accretion problems that we could consider, we focus on the dynamics of accretion flow onto a moving black hole, where we again emphasize that we assume 1 For more examples of general relativistic astrophysical applications of fluid dynamics we refer the reader to Camenzind [5], or Shapiro and Teukolsky [6]. 1 1.1. PROJECT OUTLINE that the the mass of the accreting matter is insufficient to significantly change the mass of the black hole. With this assumption, we can describe the gravitational field using a time-independent, or stationary, spacetime. Historically, the roots of the problems that we consider can be traced to the studies of Bondi and Hoyle [7], who investigated non-relativistic accretion flows onto point particles that were moving through the fluid. Extension of such studies to the general relativistic case has been made by several authors—most notably Petrich et al. [8] and Font et al. [1]. The nomenclature “Bondi–Hoyle accretion” is typically retained in these works, and we adopt that convention here. However, these previous calculations considered only purely hydrodynamical models and, as already stated, we thus extend the earlier research by including some of the effects of magnetic fields in our work. Another departure from previous research is our modelling of the fluid, in some cases, in the so-called ultrarelativistic limit. As described in more detail in Chap. 2, we study accretion flows for spacetimes describing single spherically symmetric, or single axisymmetric black holes. The remainder of this chapter is devoted to an overview of the thesis. We begin with an outline of the thesis and highlight our main results. We then proceed to brief reviews of hydrodynamics and magnetohydrodynamics, particularly in the context of general relativistic calculations We study the accretion flow in spacetimes for spherically symmetric and axisymmetric black holes, described in more detail in Chap. 2. We proceed by presenting the outline of the thesis project with accompanying results. Then we present a brief history of hydrodynamics and magnetohydrodynamics before introducing our approach to the study of accretion flows. 1.1 Project Outline The purpose of this thesis is to study the general relativistic Bondi–Hoyle accretion problem in two distinct fluid models. The first part of the study uses an ultrarelativistic fluid model, where the rest mass density of the fluid is neglected. The second part of the study generalizes the relativistic Bondi–Hoyle problem by using a background fluid which is a perfect conductor with a magnetic field embedded in it. The details for these models are found in Chap. 2. To perform this study we focus on two different fluid descriptions for the uniform fluid background used in the general relativistic Bondi–Hoyle setup: 1. we investigate an ultrarelativistic fluid; 2 1.1. PROJECT OUTLINE and separately 2. we investigate an ideal magnetohydrodynamic (MHD) fluid. We use the ultrarelativistic description to model axisymmetric accretion onto an axisymmetric black hole and we also consider ultrarelativistic infinitely thin-disk accretion onto an axisymmetric black hole. The infinitely thin-disk model, the same as studied by Font et al. [3], is a highly restricted model; however, it serves the purpose of allowing us to gain an insight into the full three dimensional simulations. During our study of the ultrarelativistic systems we found a set of parameters that permit the presence of a standing bow shock. Using this new hydrodynamic model, where we neglect the rest mass density of the fluid and the corresponding conservation law, we find the presence of both a bow shock and a tail shock. We further find that the radial location of the boundary conditions for our ultrarelativistic system must be much larger than those proposed in the previous hydrodynamic studies, especially when studying subsonic and marginally supersonic flows. We discuss a comparison between our model and previously studied hydrodynamic models in Chap. 5. When we use the ideal MHD model we study the axisymmetric accretion onto an axisymmetric black hole with an asymptotically uniform magnetic field aligned with the rotation axis. While this geometric setup is highly idealized it reveals several new physical features not seen in the previous studies of a purely hydrodynamic fluid background. One such feature is a region downstream of the black hole that evacuates, forming a vacuum. The depletion region is a phenomenon that appears to be similar to the effects of our Sun’s solar wind interacting with the earth’s magnetosphere. We also find that the presence of a magnetic field only marginally affects the accretion rates relative to a hydrodynamic system where there is no magnetic field present in the fluid. To investigate our fluid models in the relativistic Bondi–Hoyle accretion problem we focused on three distinct combinations of equations and domain geometries: 1. Axisymmetric, ultrarelativistic accretion; 2. Non-axisymmetric, infinitely-thin ultrarelativistic accretion; 3. Axisymmetric, magnetohydrodynamic accretion. We developed our own finite-volume high-resolution shock-capturing code. Since the field of numerical magnetohydrodynamics is still very new, there are a lot of advantages to developing our 3 1.2. NUMERICAL RELATIVISTIC HYDRODYNAMICS: A BRIEF REVIEW own code, including developing a much better fundamental understanding of the methods and techniques used in the field. No existing code at the time of this writing used the ultrarelativistic equations of motion to be described in Chap. 2, nor did any code exist that used our implementation of the magnetohydrodynamic equations of motion. Further to this we suggest a new method to monitor the physical validity of the magnetic field treatment, in Chap. 4. 1.2 Numerical Relativistic Hydrodynamics: A Brief Review A plasma is a highly ionized gas, that is, it is a gaseous mixture of electrons and protons. With current numerical methods and computational facilities there is no efficient way to study the dynamics of every particle in the plasma. Since we are interested in the bulk properties of plasma accretion we approximate the plasma flow using (magneto)hydrodynamic models. In this section we describe the hydrodynamic approximation, where we assume the plasma may be treated as a fluid. Since we will be using the Valencia formulation [9], and integral techniques to solve the resulting system of equations, we briefly introduce those concepts here, expanding them in greater detail in Chap. 3. We also introduce the concepts and assumptions needed for an ultrarelativistic fluid. 1.2.1 Ideal Hydrodynamic Approximation We are interested in accretion of astrophysical plasmas onto black holes, and since we are specifically interested in the gravitational attraction of the fluid to the black hole, we will be concerned with the flow of the heaviest particles. To simplify the model, we will make the assumption that the particles are all baryons which all have identical mass, mB . We will further use the hydrodynamic approximation, which means that we assume that the fluid will be adequately described by studying the bulk properties of the particles within fluid elements, or small volumes of fluid. The size of the fluid element is much larger than the mean free path of each particle constituent, and consequently each fluid element is considered to be in local thermal equilibrium. Thermal equilibrium suggests that the velocity distribution in each fluid element is isotropic. An isotropic velocity distribution further implies that the pressure the particles exert on the sides of the fluid elements is also isotropic [10]. 4 1.2. NUMERICAL RELATIVISTIC HYDRODYNAMICS: A BRIEF REVIEW Moreover, since the typical velocity of the constituent fluid particles in each fluid element will be of order of the speed of light, at least in some regions of the domain, we must consider relativistic hydrodynamics. 1.2.2 Review The numerical study of hydrodynamics has its roots in Euler’s mathematical analysis of fluid dynamics in 1755 [11]. For a review of non-relativistic hydrodynamics, we refer the reader to Darrigol (2008) [12] and Goldstein (1969) [13]. For a review of the numerical methods used to solve non-relativistic hydrodynamic equations of motion, we refer the reader to Birkhoff (1983) [11]. While numerical hydrodynamics has been applied to many physical systems, the most relevant studies for our current project relate to the development of hydrodynamical models for relativistic astrophysical systems. An early attempt to study astrophysical phenomena using a hydrodynamic model was performed by May and White [14, 15] in their study of the spherically symmetric gravitational collapse of a star. Their study used Lagrangian coordinates, wherein the coordinates move with the fluid. To discretize their system of equations, May and White used a finite difference scheme where one replaces the derivatives in an equation with approximate differences. To handle discontinuities that may develop in the fluid variables, they introduced an artificial viscosity by adding a term to the system of equations to mimic the effects of physical viscosity. The viscous term acts to smooth discontinuities so that the fluid variables may be treated as continuous. Lagrangian coordinates make it difficult to consider problems in general geometries. In spherical symmetry, the matter may only move radially and the co-moving coordinates cannot “pass” each other; however, in more general spacetimes the matter has greater degrees of freedom, which can allow the mixing of coordinates [16]. In contrast to the Lagrangian system, we can use Eulerian coordinates, where the coordinates are fixed by some external conditions and the fluids flow through the coordinate system [17]. An early relativistic investigation using Eulerian coordinates to describe a fluid system was performed by Wilson [17, 18] in the 1970’s, where he used splitting methods2 with conservative, flux-based treatment along coordinate directions, and include the use of artificial viscosity for smoothing shocks [21]. In the general form of the method used by Wilson—in the case of one 2 Splitting methods refers to the numerical methods where the solution of a system of multidimensional partial differential equation is determined by reducing the original multidimensional problem into a series of one dimensional sub-problems. For more details on splitting methods we refer the reader to [19, 20]. 5 1.2. NUMERICAL RELATIVISTIC HYDRODYNAMICS: A BRIEF REVIEW spatial dimension, x—each fluid unknown, q = q(x, t), satisfies a partial differential equation which takes the form of a so-called advection equation, ∂q ∂vq ∂q + = s q, ∂t ∂x ∂x , (1.1) where v = v(t, x) is the fluid 3-velocity. It is important to note that in this formulation the source term, s = s(t, x, ∂q/∂x) contains gradients of the fluid pressure, and thus this is not a so-called “conservation equation”. The Wilson method uses a finite difference approximation to numerically evolve the resulting partial differential equations. The use of the finite difference method also required the introduction of an artificial viscosity that would stabilize the numerical method in the presence of a discontinuity such as a shock [22]. Since this method required the use of artificial viscosity for numerical stability this method cannot highly resolve shocks without very high numerical precision. There are known limitations to the Wilson method for some physical systems including ultrarelativistic flows [21]. The limitations are attributed to the use of artificial viscosity, and consequently, for such situations the development of a different numerical method was necessary3 . The formulation that we use originated with the Valencia group [9], who realized that the fluid equations may be written as a set of coupled conservation equations. The Valencia formulation is similar to the one described by Wilson; however, the equations are now written as, ∂q + ∇ · f (q) = s ∂t (1.2) where q is a vector of conservative variables and the f are known as the fluxes associated with the conservative variables. In this description the components of the vector of source functions, s, do not contain any spatial derivatives of the fluid variables such as the velocity or the pressure. By using this formulation, Mart´ı et al. [9], were able to adopt Godunov-type schemes (also known as high resolution shock capturing, or HRSC, schemes) which will be described in detail in Chap. 3. Godunov-type schemes solve an integral formulation of the conservative equation (1.2), and are therefore also valid across discontinuities in the fluid variables. These methods do not require the use of artificial viscosity to stabilize the resulting evolution. From here on we refer to this approach 3 The Wilson method is still actively used in modern research, and has been used to investigate many different physical systems such as accretion flows, axisymmetric core collapse as well as coalescence of neutron star binaries [21]. 6 1.2. NUMERICAL RELATIVISTIC HYDRODYNAMICS: A BRIEF REVIEW as the conservative method. A generalization of the Godunov-type schemes are called finite volume methods. Finite volume methods find “weak” solutions to hyperbolic systems of equations4 and are capable of solving hyperbolic partial differential equations with discontinuous data sets. We describe this method in more detail in Chap. 3. Godunov’s original scheme [23] is only a first order accurate integral solution; however, using conservative schemes, researchers were able to develop methods that extended the numerical accuracy of the integral solution. This allows for better resolution of the extreme pressure gradients, and other discontinuities, that form in a supersonic fluid. When studying a one dimensional advective equation such as Eqn. (1.2), we refer to a characteristic curve as the path along which values of the field, q, propagates unaltered with a characteristic speed, ∂f /∂q. Mathematically, along a characteristic curve, τ (x, t), Eqn. (1.2) reduces from a partial differential equation (PDE) to an ordinary differential equation (ODE), ∂q dq + ∇ · f (q) = s ⇒ = s. ∂t dτ (1.3) The mathematical details behind characteristics are found in Chap. 3. By formulating the hydrodynamic equations in a conservative form, researchers were able to calculate the characteristics, or characteristic structure, for all the fluid variables. Moreover, since general relativity requires all physical equations of motion to be hyperbolic, due to speed of light constraints on propagation of physical effects, it certainly seems natural to consider the use of conservative methods when studying fluid systems in a general relativistic context.5 The influences of hydrodynamics on gravitational systems and vice versa has been the subject of a large amount of research for decades. Researchers have performed detailed surveys of idealized fluid systems in many different configurations, starting as early as the 1970’s when Michel studied steady state accretion onto a static spherically symmetric black hole [25]. In the 1980’s, Hawley and collaborators, studied accretion tori (donut-shaped disks with high internal energies and welldefined boundaries [26]) around rotating black holes using a relativistic hydrodynamical code based 4 A hyperbolic partial differential equation has a well-posed initial value problem, and typically represent wave-like phenomenon. Solutions to hyperbolic PDEs are wave-like if any disturbances travel with finite propagation speed. We refer the reader to [19, 20] for more details. 5 For a review of this method of study please see Living Reviews in Relativity, in particular “numerical hydrodynamics and magnetohydrodynamics in general relativity” [21]. Interested readers may also wish to read the review “numerical hydrodynamics in special relativity” [24]. 7 1.2. NUMERICAL RELATIVISTIC HYDRODYNAMICS: A BRIEF REVIEW on Wilson’s method [27, 28, 29]. This work is reviewed in Frank, King, and Raine [30]. In Hawley et al.’s setup [28], the stationary black hole is centred in an axisymmetric thick accretion disk and evolved in time. They parameterized their accretion torus by the angular momentum of the entire torus, l, and studied three regimes, one in which initially l < lms , where lms is the specific angular momentum of a marginally stable bound orbit, another for lms < l < lmb where lmb is the angular momentum of the marginally bound orbit, and finally lmb < l. For l < lms the accretion torus was found to flow into the black hole. When lms < l < lmb only some of the disk flows into the black hole while the rest remains in orbit, and finally when lmb < l the accretion torus was found to remain orbiting outside the black hole. De Villiers and Hawley extended this study [31] by considering the full three dimensional accretion tori and investigated the effect of the Papaloizou–Pringle instability, an instability found in constant specific angular momentum accretion tori when disturbed by non-axisymmetric perturbations [32]. Before we proceed, we will briefly introduce a concept related to relativistic hydrodynamics, that is ultrarelativistic hydrodynamics. This is one of two particular models that are the focus of this research. 1.2.3 Ultrarelativistic Hydrodynamics When the characteristic fluid velocities of the particles that make up the fluid elements are very close to the speed of light, the thermal energy of the fluid is much greater than the rest mass density, and we say that the fluid is ultrarelativistic. Mathematically, this allows us to consider a limit where the rest mass density of the fluid is ignored. Ultrarelativistic systems are relevant in the early universe where the ambient temperature is thought to be on the order of T ∼ 1019 GeV, and the internal energy of the particles is far too high for the rest mass density to affect the system [33]. The ultrarelativistic model of a fluid is particularly useful in the radiation-dominated phase of the universe, where we would naturally expect to find radiation fluids such as photon gases [34]. The black holes in this period would be primordial black holes [33]. The algebraic details for this fluid model will be discussed in Chap. 2. Ultrarelativistic fluids have been studied in detail for stellar collapse [35, 36, 37], and have been treated as a model for a background fluid in Bondi–Hoyle accretion for a single set of parameters modelling the fluid [38]. We will expand on using this as a background fluid in Sec. 1.4.3. 8 1.3. NUMERICAL RELATIVISTIC MAGNETOHYDRODYNAMICS: A REVIEW 1.3 Numerical Relativistic Magnetohydrodynamics: A Review We now introduce the material needed for the second part of the thesis, relativistic magnetohydrodynamics. In particular we will focus on the assumptions behind ideal magnetohydrodynamics, as well as its history. 1.3.1 Ideal Magnetohydrodynamic Approximation We extend the ideal hydrodynamic approximation, that the accreting plasma may be treated as a single constituent fluid, by imposing the ideal MHD limit. We assume that the fluid is a perfect conductor which imposes the condition, via Ohm’s law, that the electric field in the fluid’s reference frame vanishes, and that the electromagnetic contributions to the fluid are entirely specified by the magnetic field. This is shown in mathematical detail in Chap. 2. The perfect conductivity condition leads to “flux-freezing”, where the number of magnetic flux lines in each co-moving fluid element is constant in time. 1.3.2 Review Having introduced the development of the numerical techniques to solve hydrodynamic models for astrophysical phenomenon, we turn our attention to a review of the material where researchers extend the existing hydrodynamic numerical techniques to include magnetic field effects. In particular, we focus on the case where they take the ideal magnetohydrodynamic limit, so the fluid is treated as a perfect conductor therein no electric fields are present in the fluid’s reference frame. This is discussed in greater detail in Chap. 2. Both the Wilson formulation, Eqn. (1.1), and the conservative method, Eqn. (1.2), were extended to magnetic-fluids in the ideal magnetohydrodynamic limit in the early 2000’s. De Villiers and Hawley (2003) extended the Wilson formulation to include magnetic fields in De Villiers et al. [39], where they studied accretion tori on Kerr black holes. Hawley et al. studied the hydrodynamic accretion tori in Hawley et al. (1984) [28] and added a weak poloidal magnetic field to trigger a magnetorotational instability (MRI) [26]. The flows resulted in unstable tori in which the MRI develops and is later physically suppressed due to the symmetry of the setup. Anninos et al. (2005) [40] extended De Villier’s work by studying a variation of the Wilson technique when calculating 9 1.3. NUMERICAL RELATIVISTIC MAGNETOHYDRODYNAMICS: A REVIEW stable magnetic field waves. The magnetohydrodynamic extension of the Wilson method has also been used to study different accretion phenomenon such as tilted accretion disks [41]. As with the original Wilson formulation, the extensions were finite-difference based codes and therefore also required the introduction of artificial viscosity to smooth discontinuities. All general relativistic magnetohydrodynamic codes are based on developments made in special relativistic magnetohydrodynamics. Notable special relativistic developments include work by Van Putten (1993) [42], who used a spectral decomposition code to solve the equations of motion. He proved the existence of compound waves in relativistic MHD, analogous to the magnetosonic waves found in classical MHD by Brio and Wu [43].6 Later Van Putten (1995) [45] calculated the fully general relativistic ideal magnetohydrodynamic equations of motion. Both studies by Van Putten required the use of smoothing operators to stabilize shocks, and consequently were not able to accurately handle problems that contained high Lorentz factors. This is due to the “smearing” caused by the smoothing operators which substantially reduce the accuracy of solutions across strong shocks [46]. Balsara (2001) [47], was the first to calculate the closed form analytic solution of the special relativistic characteristic structure and used a total variation diminishing (TVD) Godunov-type scheme to solve the 1-dimensional magnetohydrodynamic equations of motion. Komissarov (1999) [46] was the first to develop a 2-dimensional second order Godunov-type special relativistic MHD solver. Then del Zanna et al. (2003) [48] become the first group to develop a 3-dimensional third order Godunov-type scheme for special relativistic MHD. Although the implementation details vary from code to code, such as the use of spectral methods, TVD methods, and higher order schemes, all are based on conservative formulations of the fluid equations of motion. The Godunov-type scheme requires that we solve a Riemann problem either exactly or approximately. The Riemann problem consists of a conservation law in conjunction with piecewise constant initial data that contains a single discontinuity. The Riemann problem is the simplest model for discontinuous systems. Mart´ı and M¨ uller (1994) [49] were the first to develop an exact Riemann solver for 1-dimensional relativistic hydrodynamics, later extended to multiple dimensions by Pons et al. (2000) [50]. Rezzolla and Giacomazzo (2001) refined this method [51] and later extended it to special relativistic magnetohydrodynamics (2006) [52]. The exact solution is useful in code testing and verification of the different flux approximations that are used for the approximate Riemann solvers. No exact solution exists for a Riemann problem in general relativistic magne6 We do not introduce spectral methods in this thesis but will refer the reader to textbooks on the subject such as “Spectral Methods for Time-Dependent Problems” [44], as an example. 10 1.3. NUMERICAL RELATIVISTIC MAGNETOHYDRODYNAMICS: A REVIEW tohydrodynamics, but since the Riemann solvers are local, we can use special relativistic exact solvers if we change to the appropriate reference frame [53]. The Godunov-type methods described in the previous paragraph use approximate Riemann solvers, where the iterative processes used to solve the exact Riemann problem are replaced with approximations that are faster to solve. These approximate methods are discussed in Chap. 3. With the special relativistic conservative equations of motion for ideal magnetohydrodynamics developed, work began by Anile [54] on the construction of Godunov-type conservative methods that could include the relativistic equations of motion on a curved spacetime background. Gammie et al. (2003) [55] were the first to use this method to develop a general relativistic code called HARM (High-Accuracy Relativistic Magnetohydrodynamics). Komissarov (2005) [56] used conservative methods to describe the magnetosphere of a black hole. Anton (2006) followed up on Mart´ı’s 1991 [9] paper by investigating and subsequently calculating the “characteristic-structure” of magnetohydrodynamics in a general relativistic fixed background [57]. The accretion torus problem was studied in the context of a magnetohydrodynamic fluid accretion using conservative methods by Gammie [55], and Montero with Rezzolla [58], which also resulted in a simulation of the MRI. For a review of the magnetized torus problem we refer the reader to papers such as de Villiers et al. (2003) [39]. Now that we have reviewed the development of the techniques used to study general relativistic magnetohydrodynamic (GRMHD) systems with a fixed spacetime background7, we turn our attention to the applications of GRMHD to astrophysical problems, particularly the accretion process. When material accretes onto a massive central object the material will begin to reduce its orbital radius. If the material has angular velocity relative to the central object, to conserve angular momentum, the angular velocity will increase as the radius is reduced. In the most extreme cases, such as accretion onto compact objects, the velocity of the accreting material will approach the speed of light. At these limits, if the material were to reduce its orbital radius any further, the corresponding increase in angular velocity would exceed the speed of light. Consequently such material would cease to accrete onto the central object, thus researchers were led to ask questions about how the angular velocity or angular momentum would be transported away from the material closest to the central object. In typical fluids found on earth, one may expect that frictional 7 There are several other applications and techniques for both hydrodynamic and magnetohydrodynamic matter models including the simulation of core-collapse supernovae and neutron star mergers. As these depend on treating a dynamic spacetime background, and do not relate directly to the thesis topic, I do not discuss them here. Reviews of these topics may be found in papers such as [59], [60], [61] or [62]. 11 1.3. NUMERICAL RELATIVISTIC MAGNETOHYDRODYNAMICS: A REVIEW forces such as viscosity would allow for this mechanism to take place. In large scale astrophysical accretion processes; however, viscosity is too small to be the dominant mechanism for angular momentum transport [26]. Researchers such as Stone, Balbus and Hawley [63, 64] investigated angular momentum transport and found that for steady hydrodynamic flow, there cannot be any angular momentum transport out of the disk, and further that if an instability is present, the angular momentum transport must be inward [26]. Balbus and Hawley (1998) reviewed accretion as well as the effects of magnetic fields in accretion phenomena [26]. In the review they show that a hydrodynamic description of an accretion disk is not capable of allowing angular momentum transport, all explanations using hydrodynamic models such as differential rotation are stable to linear perturbations. Balbus and Hawley encouraged further exploration of the role of magnetic fields in accretion, since introducing a magnetic field to the accretion system introduces instabilities that do not exist in a purely hydrodynamic system. When the accreting fluid contains an embedded magnetic field, Balbus and Hawley show that the accretion disk experiences the so-called magnetorotational instability [65]. Hawley et al. [66] explain that the viscous dissipation from magnetic field can come from two possible torques, either external, where a rotating magnetized wind coming off the disk carries away angular momentum, or internal, where the magnetic fields carry the angular momentum radially out of the disk by a linear instability in the disk due to an angular momentum transfer process in the presence of a weak magnetic field. The existence of a rotational velocity in the magnetohydrodynamic system allows for an incompressible magnetorotational wave which is unstable for some wavenumbers. The magnetorotational instability is caused by the magnetic tension transferring angular momentum from fluid elements in low orbits, with large angular velocity, to fluid elements in higher orbits and smaller angular velocity. To conserve angular momentum, an object accelerated in the direction of its orbit that gains angular momentum moves to a higher orbit, thereby decreasing its angular velocity. The magnetic fields enforce co-rotation between these fluid elements and ultimately decelerate the fluid element in the lower orbit and accelerate the fluid element in the higher orbit. This process transfers angular moment away from the accreting body. This effect is only possible for weak magnetic fields, otherwise the magnetic field force dominates any centrifugal force of the fluid and holds the fluid together [66]. As general relativistic magnetohydrodynamics is a rapidly developing field, there are many outstanding questions, including how magnetic fields affect the accretion rates found in systems 12 1.3. NUMERICAL RELATIVISTIC MAGNETOHYDRODYNAMICS: A REVIEW that were previously studied using the purely hydrodynamic approximation. One of the models used to explain a particular type of accretion phenomenon is referred to as Bondi–Hoyle accretion. Bondi–Hoyle accretion is thought to be a good model for accretion inside common envelopes, astrophysical bodies in a stellar wind, and bodies inside active galactic nuclei [67]. We briefly describe these below and we refer the reader to texts such as “Introduction to High-Energy Astrophysics” [68] for more details. In a close binary system mass can transfer from one object to another. When the object transfers mass, it also transfers momentum, and therefore causes a change in the orbital separation [68]. If mass is transfered from an object of lower mass to an object of higher mass, the lower mass object moves in such a way as to increases the orbital separation between the binary bodies so that it conserves the linear momentum of the entire system and angular momentum of the orbiting body. On the contrary, if a larger mass object transfers mass to a smaller mass object, the orbital separation will decrease. If the latter occurs, an unstable mass transfer may ensue [68]. One such outcome will be a common envelope. When mass is transfered from the more massive donor star to the less massive accreting star or black hole, such that the mass transfer is faster than the accretor may accrete it, a hot cloud, or envelope, of stellar matter forms around the accretor. If the envelope grows large enough it will become larger than the size of the Roche lobe, the region around a star in which orbiting material is gravitationally bound to the star. It will therefore envelop both stars, becoming what is known as a common envelope (CE). The CE will then exert a drag force on the orbiting bodies, which will reduce the orbital radius of the binary system. The energy extracted from the binary stars is deposited in the common envelop as thermal energy [68]. Stellar wind is the emission of particles from the upper atmosphere of a star [69]. The amount of matter that makes up the stellar wind will depend on the star producing the wind. Dying stars produce the most stellar wind, but this wind is relatively slow at ∼ 400 km s−1 [70], while younger stars eject less matter but at higher velocities, ∼ 1500 km s−1 [71]. Bondi–Hoyle type accretion occurs when a massive object passes through this material [67]. An active galaxy is a galaxy in which a significant fraction of the electromagnetic energy output is not contributed by stars or interstellar gas. At the centre of the active galaxy lies the nucleus, commonly known as an active galactic nucleus (AGN), which is on the order of 10 light years in diameter [68]. The radiation from the core of an active galaxy is thought to be due to accretion by 13 1.4. BONDI–HOYLE ACCRETION a super massive black hole at the core, and generates the most luminous sources of electromagnetic radiation in the Universe. One of the modern applications of Bondi–Hoyle accretion may be seen in Farris et al. [72], where they simulate the merger of two binary black holes in a (non-magnetic) fluid background. They simulate both a Bondi-like evolution where the background gas is stationary relative to the binary merger, as well as a Bondi–Hoyle evolution where there is a net velocity of the fluid background relative to the binary merger. All existing Bondi–Hoyle accretion models have focused on the purely hydrodynamic case, despite the fact that the phenomenon described above may be treated in a more general sense by allowing for the presence of an electromagnetic field [73]. In this thesis, we address two distinct physical scenarios. First, we study the accretion of a truly ultrarelativistic hydrodynamic fluid onto a black hole in a Bondi–Hoyle model in two distinct geometric configurations: axisymmetric and non-axisymmetric infinitely thin-disks, which are described in more detail in Chap. 2. Second, we investigate Bondi–Hoyle type accretion onto a black hole using an axisymmetric magnetohydrodynamic model. In all of our studies, we are interested in looking for phenomenological effects such as instabilities that may develop, or if the flow reaches a steady state. To determine if the flow is stable, we will measure the accretion rates of energy, mass, and angular momentum. In the event of a stable accretion flow, the accretion rates will be constant in time. We now describe the Bondi–Hoyle accretion model as well as summarizing the history of the studies performed using this model. 1.4 Bondi–Hoyle Accretion The Bondi–Hoyle accretion problem, whether it is considered in the gravitationally non-relativistic regime or in the relativistic regime, has the same basic setup. A star, of mass M , moves through a uniform fluid background at a fixed velocity, v∞ , as viewed by an asymptotic observer. We assume that the mass accretion rate is insufficient to significantly alter the spacetime background around the accretor [67]. Likewise, any momentum accretion rates are assumed too small to alter the velocity of the central body. We now give a brief history of the study of the Bondi–Hoyle system in the non-relativistic and relativistic regimes. 14 1.4. BONDI–HOYLE ACCRETION v∞ b θ M Figure 1.1: The original Hoyle–Lyttleton geometric setup. A particle with uniform asymptotic velocity v∞ with an impact parameter b travels towards the massive body, M . As it passes the point mass the trajectory is altered, and if the particle is close enough to the point mass the particle will converge onto the x-axis, as shown by the solid black line. b t Figure 1.2: The original Bondi–Hoyle geometric setup. This setup is identical to the Hoyle– Lyttleton approach with the exception that the accretion column, the region labelled ’t’, that forms behind the point mass is also considered. By including the accretion column the Hoyle–Lyttleton accretion rate was found to reduce by half. 15 1.4. BONDI–HOYLE ACCRETION For the sake of this review of previous work, when we refer to a non-relativistic fluid, the characteristic velocities of the (magneto)hydrodynamic system is sub-relativistic and the gravity is treated in a Newtonian framework. When a fluid is relativistic, it has characteristic velocities that approach the speed of light, is modelled using either special or general relativity, and the gravity is treated using a general relativistic framework. Our research does not focus on the nonrelativistic Bondi–Hoyle accretion problem; however, it is important to understand the features of a non-relativistic system, since many of these same features appear in the relativistic regime. 1.4.1 Non-relativistic Regime To understand the non-relativistic Bondi–Hoyle accretion problem as it was originally posed, we follow the ballistic trajectory of a streamline as the fluid passes the point mass. The trajectory of the fluid will be affected by the gravitational field of the star. If the kinetic energy of the flow is smaller than the gravitational energy of the mass the trajectory of the flow will be altered and the fluid will ultimately be accreted by the mass. As the fluid caught by the gravitational field passes the star, it becomes caught in the gravitational field of the star. This results in the fluid flow changing its trajectory and converging behind the star. If the fluid is too close to the star the fluid will be gravitationally bound to the star, reverse direction and accrete onto the star [67]. Using the assumption of a ballistic trajectory, we neglect the effects due to pressure within the fluid. The study performed by Hoyle and Lyttleton (1939) [74] determined the rate of accretion of a massive star as it travels through a uniform pressureless fluid background. A schematic representation is found in Fig. 1.1. Their proposed closed form analytic model of the mass accretion rate, M˙ HL , was; 4πG2 M 2 ρ∞ . M˙ HL = 3 v∞ (1.4) This is known as the Hoyle–Lyttleton accretion rate, M˙ HL . Here, G is Newton’s gravitational constant, while M and v∞ are the mass and the velocity of the massive point-like object as viewed by an asymptotic observer. Likewise, ρ∞ is the density of the fluid as viewed by an observer at infinity. For the derivation we refer the reader to references [74, 75, 76, 77, 67]. Bondi and Hoyle then re-investigated the original Hoyle–Lyttleton system but changed the geometry to reflect the presence of the accretion column seen in Fig. 1.2 [7, 67]. By including the accretion column they found that the Hoyle–Lyttleton accretion rate in Eqn. (1.4) was reduced by 16 1.4. BONDI–HOYLE ACCRETION a half; producing the Bondi–Hoyle accretion rate, M˙ BH , 2πG2 M 2 ρ∞ M˙ BH = . 3 v∞ (1.5) Several years later Bondi [78] studied the mass accretion rate onto a spherically symmetric point mass he calculated a mass accretion rate for an object with zero velocity relative to the fluid background with a non-zero pressure, G2 M 2 ρ∞ M˙ B = 2π 3 , (c∞ s ) (1.6) where c∞ s is the speed of sound in the fluid as viewed by an observer at infinity. Due to the similarities between Eqn. (1.5) and Eqn. (1.6), Bondi posited an interpolation formula to connect the mass accretion rate of a point-like body with a fixed velocity to the spherically symmetric accretion problem, formulating an accretion rate M˙ B [67, 78], M˙ B = 2π G2 M 2 ρ∞ 3/2 2 + v2 ) (c∞ s ∞ . (1.7) By performing a series of numerical studies of the Bondi–Hoyle and Hoyle–Lyttleton accretion Shima et al. [79] found that the correct scaling factor for Eqn. (1.7) is twice as large as Bondi’s original calculation. The final non-relativistic mass accretion rate was determined to be, M˙ B = 4π G2 M 2 ρ∞ 3/2 2 + v2 ) (c∞ s ∞ . (1.8) Extensive numerical work was carried out, using the non-relativistic treatment of both the fluid and the gravitational field, by Ruffert in the mid-nineties [80, 81, 82, 83, 84, 67] and, more recently, in 2005 [85], where he found that the three dimensional evolution was stable. In the non-relativistic 2D simulations, the massive body travels through space with sub-relativistic speeds. As this body travels, it begins to create an accretion column in its wake. The pressure in the column builds as the flow evolves, in doing so the accretion column widens. If equilibrium is reached between the pressure in the accretion column and the oncoming fluid the tail shock will stabilize, and stop widening. If equilibrium is not reached the column will continue to grow moving closer to the upstream region until it loses contact with the massive body, forming a bow shock. In our context, 17 1.4. BONDI–HOYLE ACCRETION a bow shock8 is a curved shock wave that forms in front of a massive body as the fluid flows past the body. This is in contrast to a tail shock9 which is a shock wave that forms behind the massive body starting at the body and extending outward. In several simulations, just as the bow shock was forming, the point of contact between the accretion column and the massive body would oscillate from side-to-side around the massive body, eventually wrapping around the body destabilizing the flow. This oscillation was called the “flip-flop instability”. Later it was discovered that this instability was actually very sensitive to boundary conditions in each simulation. Some, such as Foglizzo et al. (2005) [86], have argued that the flip-flop instability is an artifact of an incorrect numerical treatment of the boundary conditions, while others e.g. Blondin (2009) [87], have recently argued that this instability is real. As stated in Font et al. (1998) [2], the 3D simulations by Ruffert (1999) [88] show strong evidence of long term stability of the accretion flows, unlike the simulations performed in two spatial dimensions studied in Sawada (1989) [89], Matsuda (1991) [90] and Livio (1991) [91], unless the flow has density gradients at infinity [92]. Since our work is in the relativistic regime, we do not address the issue of the presence of a flip-flip instability in non-relativistic Bondi–Hoyle accretion. In the non-relativistic system, a bow shock develops depending on the values of the parameters used to specify the flow:10 1. the adiabatic constant, Γ for polytropic equations of state; 2. the asymptotic speed of sound in the fluid, c∞ s ; 3. the Mach number, M = v/cs , where v is the speed of the accretor; 4. and the radius of the accretor, ra . Ruffert discovered that a bow shock develops in these systems due to a pressure increase in the downstream flow. As the body moves through the background fluid, it will begin to attract material and start compressing the material closest to the upstream side of the accretor. Due to the compression of the upstream matter the internal pressure and density increase, high enough that a shock forms. Studies have shown that the position of the shock is controlled by the value of Γ, and not by the Mach number, M. 8A bow shock is also commonly referred to as a detached shock. tail shock may be referred to as an attached shock. 10 Precise definitions of the adiabatic constant, speed of sound, and Mach number are given in Chap. 2. 9A 18 1.4. BONDI–HOYLE ACCRETION We note that non-relativistic Bondi–Hoyle accretion is still under investigation in various configurations, including configurations where the composition of the background fluid is no longer uniform [93, 94]. 1.4.2 Relativistic Regime Michel (1972) [25], studied a relativistic extension to the Bondi accretion problem, where a uniform fluid surrounds a stationary spherically symmetric black hole, with the assumption that the background fluid is not sufficiently massive to modify the mass of the black hole. He discovered that the relativistic Bondi problem has a closed form solution, depending on two free parameters which are set by specifying the density and pressure at a transonic point, the point where the fluid velocity equals the speed of sound. This solution is explained in more detail in Chap. 4. The relativistic Bondi–Hoyle accretion problem consists of a black hole travelling at a constant asymptotic velocity through a uniform fluid background. In analogy to the non-relativistic Bondi– Hoyle problem, we assume that the mass of the accreted matter is small relative to the mass of the black hole. This allows us to examine the system using a fixed spacetime background. To date no closed form solution to this problem has been found. The numerical study of the relativistic Bondi–Hoyle accretion problem was originally performed by Petrich et al. (1989) [8]. They numerically studied axisymmetric accretion onto a spherically symmetric black hole using the Wilson method. The fluid was completely specified by the velocity, v∞ , of the travelling black hole viewed by an asymptotic observer, by the adiabatic constant, Γ, used in the equation of state, and by the asymptotic speed of sound, c∞ s , for the fluid. Petrich et al. [8] showed that the evolution of such a system settles down to a steady state flow onto the black hole. In the same study, Petrich et al. proceeded to calculate the accretion rates for both the mass and angular momentum relative to the relativistic Bondi mass accretion rate (1.7). They discovered a good agreement between the Bondi–Hoyle theory and their non-relativistic evolution calculations. Their paper also surveyed a wide range of parameters, from Newtonian limits up to what they call the ultrarelativistic limit, where the speed of sound and the speed of the black hole are close to the speed of light. The use of a time dependent code allowed them to determine that the resulting steady state flow was both unique and stable [8]. In later works, Font, Iba˜ n´ez, and Papadopoulos [1, 2, 3, 95] used conservative methods to solve the hydrodynamic equations of motion, and re-investigated the relativistic Bondi–Hoyle setup. 19 1.5. THESIS LAYOUT In Font et al. (1998) [1], all simulations performed on a wide range of values for M, c∞ s and Γ resulted in a steady evolution, and all but one of their simulations produce a tail shock. In Font et al. (1998) [2], they changed their geometric configuration to calculate the solution of nonaxisymmetric hydrodynamic accretion onto the equatorial plane of a spherically symmetric black hole using a thin disk approximation. The results of [2] were again steady state solutions, each with a tail shock. Finally, in Font et al. (1999) [3], they studied non-axisymmetric accretion onto a rotating black hole using an infinitely thin-disk approximation where they discovered that these flows also reached a steady state with an attached tail shock, but also that the effects of the rotating black hole were only noticeable within a few Schwarzschild radii, Rs from the black hole; beyond that the flow was essentially unaltered by the rotation. The relativistic studies described above did not find the presence of the “flip-flop” instability found in the non-relativistic accretion models. 1.4.3 Ultrarelativistic Fluid Modelling Past research using ultrarelativistic fluid models mainly focused on gravitational collapse problems. There has been little research into the accretion of an ultrarelativistic fluid onto a compact object itself. Petrich et al. [38] studied the special case of ultrarelativistic accretion onto a spherically symmetric compact object for a Γ = 2 fluid. They found that the problem could be solved in closed form, and that the flow was steady. Polytropic fluids with Γ = 2 have a speed of sound equal to the speed of light, therefore they are all subsonic. One of our goals is to see what happens as the adiabatic constant decreases and the fluid is allowed to become supersonic. Only numerical treatments appear to be able to solve the complicated system of partial differential equations that arise from using fluid models. 1.5 Thesis Layout In Chap. 2, we describe the necessary general relativistic formalism to describe spacetimes. We describe the coordinates used for this study and the stress-energy tensors used to calculate the equations of motion of the fluid for this work. We derive the equations of motion for both the magnetohydrodynamic models and the ultrarelativistic hydrodynamic models. We also describe the specific geometries of the spacetimes which we considered. In Chap. 3, we describe our Godunov-type solver and specifically the finite volume method, used 20 1.5. THESIS LAYOUT to numerically solve a system of hyperbolic partial differential equations. We describe the different flux approximations we implemented, along with the different cell interface techniques used. We further describe techniques both for handling discontinuities that arise in the fluid evolution, known as shock capturing, and for maintaining the magnetic field constraint to truncation error. In Chap. 4, we describe a set of tests that are used to determine the validity of the flux approximations and reconstruction techniques used to measure the shock capturing capabilities of the methods described in chapter 3. This includes one and two dimensional tests. We also describe methods used for code verification, in particular the convergence test, and the independent residual, useful for simulations with no known solution for comparison, since they allow us to determine whether or not our numerical solution approaches the continuum solution. For the general relativistic magnetohydrodynamic study, we suggest a new method to study the convergence of the magnetic field constraint. In Chap. 5, we present the results of our current work and we describe the simulations using both axisymmetry and thin-disk approximations for both the ultrarelativistic and magnetohydrodynamic systems. We further go on to describe the results of simulations of axisymmetric fluid flow onto a rotating black hole which include the following simulations: 1. Axisymmetric ultrarelativistic accretion onto an a = 0 black hole; 2. Axisymmetric ultrarelativistic accretion onto an a = 0 black hole; 3. Thin-disk ultrarelativistic accretion onto an a = 0 black hole; 4. Thin-disk ultrarelativistic accretion onto an a = 0 black hole; 5. Axisymmetric magnetohydrodynamic accretion onto an a = 0 black hole; 6. Axisymmetric magnetohydrodynamic accretion onto an a = 0 black hole. We found that the ultrarelativistic accretion problem in both the axisymmetry and nonaxisymmetric infinitely thin-disk models were sensitive to the location of the outer boundary. This is in contrast to previous studies. We did find in our axisymmetric studies that, once the boundary was extended far enough to prevent boundary effects from disrupting the system, some parameter combinations revealed the presence of a bow shock. In previous relativistic hydrodynamic studies only tail shocks would form for any parameters investigated. The ultrarelativistic 21 1.6. NOTATION, CONVENTIONS AND UNITS non-axisymmetric infinitely thin-disk models revealed the presence of both a tail shock and a bow shock, again a feature not discovered in previous relativistic studies. We also discovered that the presence of a magnetic field in a perfect conducting fluid background did not disrupt the development of a steady accretion flow. Furthermore, new morphological features developed in the flow, including a region immediately downstream of the black hole where the baryon rest mass density and thermal pressure are depleted. This new feature is not possible in a purely hydrodynamic system, so previous studies would not have captured them. Finally, in Chap. 6 we conclude this work and discuss related future projects. 1.6 Notation, Conventions and Units We use the following throughout the thesis: • We used MTW [4] conventions: specifically, Newton’s gravitational constant, G, and the speed of light, c are set to unity. G = c = 1. The metric signature is (− + ++). Greek indices i.e. µ, ν = {0, 1, 2, 3} run over the full spacetime coordinates, while Latin indices i.e. i, j, k = {1, 2, 3} run over the spatial components. The Einstein summation convention applies to both types of indices i.e. v µ vµ = 3 µ=0 v µ vµ and wi wi = 3 i=1 wi wi . • Since we always use a coordinate basis for calculations, we will use a standard tensor notation such that T µν will represent the components of the tensor. For ease of representation we will use the Penrose abstract index notation [96] and refer to v µ as a vector rather than the components of a vector. • We reserve the variable p to be the index which enumerates the equations in a system of conservation laws. i.e. p = 1, 2 . . . 8, 9 for the full general relativistic magnetohydrodynamic system of equations. • We denote partial differentiation by ∂i f ≡ ∂f /∂xi as well as f,i ≡ ∂f /∂xi for a function f = f (xi ). • In the case of describing a discretized quantity we adhere to the notation that a superscript n denotes a discretized time and a subscript denotes a discretized spatial location. Thus qijk ∼ q(tn , xi , yj , zk ). 22 1.6. NOTATION, CONVENTIONS AND UNITS • A dot over a variable will denote a derivative with respect to time. For example the mass accretion rate is denoted ∂m(t)/∂t = m. ˙ • We will use the acronym UHD to refer to ultrarelativistic hydrodynamics. • We will use the acronym MHD to refer to magnetohydrodynamics. • The acronym GRMHD has been used in the literature to refer to both fully dynamic general relativistic magnetohydrodynamics as well as relativistic systems in which the spacetime is fixed. In this dissertation we use GRMHD to refer to the latter. 23 CHAPTER 2 FORMALISM AND EQUATIONS OF MOTION 2.1 Overview Most of the visible Universe is a plasma, a highly ionized gas (fluid). The solution of the plasma equations of motion, as seen for example in Kulsrud [97], requires us to keep track of too many degrees of freedom for the problem to be tractable. To make the problem tractable, we usually make several assumptions about the properties of the plasma. The first assumption is that the fluid is only comprised of equal mass particles, as described in Chap. 1. Next, we are interested in studying the bulk properties of the fluid, the 3-velocity, v i , the density, ρ0 , the pressure, P , and the magnetic field B i defined in Eq. (2.77). To study the bulk properties we use the hydrodynamic approximation, allowing the bulk fluid properties to be measured by considering finite volumes or elements of the fluid. All particles within the finite element are assumed to be in local thermodynamic equilibrium with isotropic velocities. The isotropic velocity assumption leads to an isotropic pressure within each fluid element. We make this system more realistic by considering the dynamics of the fluid in the presence of a magnetic field, through the study of MHD. In this thesis we are interested in an idealized version of the MHD system, wherein the fluid body has perfect conductivity, and so the local electric field is zero. The bulk property variables are known as primitive variables and may be recorded as p = (ρ0 , v i , P, B i ). However, when we cast this system of equations in conservative form we consider the relativistic rest mass density D, the energy, E, and the 3-momentum Si . We refer to these variables as the set of conservative variables and for convenience we record them in a compact form as q = (D, Si , E, B i ). Due to the highly nonlinear nature of the relationship between the primitive and conservative variables we are not able to easily write an expression for the primitive variables, p, as a function of the conservative variables, q. The inversion is performed numerically, and is discussed in section 3.3. We will also study what is known as the ultrarelativistic limit of a hydrodynamic fluid, where 24 2.1. OVERVIEW the fluid elements described above have an internal energy that is sufficiently high that the rest mass density of the baryonic particles is negligible. So, the internal energy of the fluid which is generally written, ρ = ρ0 + ρ0 ǫ, (2.1) is dominated by the internal energy density, thus the internal energy of the fluid becomes ρ = ρ0 ǫ. (2.2) We apply the study of both ideal MHD and ultrarelativistic hydrodynamics to accretion onto black holes. Relativistic fluids are known to develop shocks. In order to capture these shocks we use highresolution shock-capturing methods to evolve the equations of motion. This technique requires the equations of motion to be written in a conservative form, ∂t q + ∂i f i (q) = s(q), (2.3) q = (q1 (xµ ), q2 (xµ ), ..., q( xµ )) (2.4) f i = (f1i (q, xµ ), f2i (q, xµ ), ..., fpi (q, xµ )) (2.5) where is a vector of conserved variables, is the corresponding flux vector, and, s(q) = s(q(xµ )) (2.6) is a vector of source terms. This will be described in more detail in Chap. 3. We will be deriving the equations of motion to take on the conservative form. We describe the spacetime defined by the presence of a black hole using Einstein’s general relativity. Einstein’s theory of general relativity is a geometric theory of gravitation. This theory is most naturally and elegantly described in a covariant form in terms of geometric objects, or 25 2.2. 3+1 DECOMPOSITION tensors. However, to study a particular problem, such as some astrophysical phenomenon, we use a coordinate dependent form; consequently, we will discuss general relativity in terms of some coordinates xµ , where µ is an index running over time and the space coordinates, as described in Chap. 1. Numerical relativity is concerned with the solution to the Cauchy problem, where a problem is given by some initial data and some boundary conditions to produce a unique time evolving solution. The Einstein equations do not lend themselves naturally to this type of problem, since they do not naturally separate space and time. Arnowitt, Deser, and Misner (ADM) [98] proposed a formulation for general relativity to allow for the solution of the Cauchy problem. Since we assume a stationary spacetime, we are not interested in the description and equations for a dynamic spacetime. We will, however, make use of the AMD variables, lapse, α, shift vector βi , and the 3-metric γij which will be defined in Sec. 2.2. In this chapter we describe the spacetimes used to model black holes and derive the equations of motion for two general relativistic fluid systems on a curved spacetime background 1. Ideal Magnetohydrodynamics Sec. 2.5, 2. Ultrarelativistic hydrodynamics Sec. 2.7. In the last section of this chapter, we discuss the different geometries studied in this project. 2.2 3+1 Decomposition We give an overview of the above mentioned ADM formalism here. For a detailed treatment please see York [99], Choptuik [100], Camenzind [5], or Alcubierre [101]. The full 3 + 1 decomposition provides us with a formalism through which we may describe the evolution of the spacetime. In our study, we consider fixed though generally curved spacetime backgrounds, so it is sufficient to restrict attention to the decomposition of the spacetime metric gµν . Assume we have a spacetime manifold and an associated metric used to measure distances between points on this manifold. We slice this spacetime into a foliation of spacelike hypersurfaces, denoted Σt , as seen in Fig. 2.1. This slicing defines surfaces of constant time, and on each surface we have three coordinates labeling spatial locations on each surface. This defines a timelike vector quantity, nν , that allows us to connect the different hypersurfaces. 26 2.2. 3+1 DECOMPOSITION Σ(t + dt) (t + dt, xi ) P′ P ′′ (t + dt, xi + dxi ) Q β idt tµ αdt Σ(t) P (t, xi ) Figure 2.1: The 3+1 decomposition of relativistic spacetime. The spacetime is foliated into space-like hypersurfaces Σ(t). Here we show two such slices at t and t + dt. The vector connecting the coordinate xi on slice Σ(t) to xi on slice Σ(t + dt) may be decomposed into a part orthogonal to the slice αdt, and a part parallel to the slice β i dt. To follow the dynamics of an event as it moves from event P at (t, xi ) to a new point Q at (t + dt, xi + dxi ) requires that we also consider the spatial translation. 27 2.2. 3+1 DECOMPOSITION If t is the time function, the normalized gradient, nµ = ∇µ t µν g ∇µ t∇ν t (2.7) is an orthogonal unit vector and ⊥µ ν , is a projection operator into the surface, ⊥µ ν ≡ δ µ ν + nµ nν . (2.8) Here δ µ ν is the Kronecker delta, and the nµ , nν = gµν nν are the normal vectors, and one-forms to the hypersurfaces respectively. i.e. if tµ is any tangent vector, then ⊥ν µ tµ is a projected tangent vector to a curve lying on the surface t = const. The induced spatial metric on the hypersurfaces is found by applying, Eqn. (2.8), to the spacetime metric, γµν = gµν + nµ nν . (2.9) The curve defined by constant spatial coordinates and varying t has a tangent vector tµ which may be written as, tµ = αnµ + β µ , (2.10) as seen in Fig. 2.1 connecting the event P at (t, xi ) to the event P ′′ at (t+dt, xi ). Here β µ = (0, β i ). The general spacetime metric may be written as; ds2 = −α2 + β i βi dt2 + 2βi dxi dt + γij dxi dxj , (2.11) where γij are the spatial components of the projected 4-tensor as defined in Eqn. (2.9). For fixed spacetime backgrounds, such as those discussed in the next section, we write the metric in the 3 + 1 formalism as follows: gtt gti gjt βs β s − α2 = gij βi βj ; γij 28 2.3. BLACK HOLE SPACETIMES and tt g g ti g jt g ij 2 −1/α = β i /α2 j 2 β /α γ ij − β i β j /α2 . We also express the determinant of the 4-metric as g, and the 3-metric γij as γ. g and γ are √ √ related by −g = α γ. The normal vector from the perspective of stationary observers on the hypersurface are: nµ = (−α, 0, 0, 0), (2.12) nµ = (1/α, −β i /α), (2.13) where nµ is defined so that we have the normalization condition nµ nµ = −1. 2.3 Black Hole Spacetimes Below we describe the coordinates used for our different spacetime backgrounds. We start with a description of a special relativistic spacetime using Cartesian coordinates, then extend the discussion to curved spacetime backgrounds including both a spherically symmetric spacetime and then an axisymmetric spacetime background. 2.3.1 Minkowski or Special Relativistic Spacetime The Minkowski metric, the standard metric used in special relativity, is diagonal, with zero shift, and has a trivial lapse function when Cartesian coordinates are used. Explicitly, the line element, ds2 , of this spacetime takes the form, ds2 = −dt2 + dx2 + dy 2 + dz 2 . (2.14) gµν = ηµν = diag(−1, 1, 1, 1). (2.15) The signature of the metric is, This description of spacetime is used to test the development of the methods described in Chap. 3. The results of some of these tests are found in Chap. 4. When we have a black hole present in our 29 2.3. BLACK HOLE SPACETIMES spacetime we consider two specific geometries. The first of these is spherically symmetric, while the second is axisymmetric. 2.3.2 Spherically Symmetric Spacetime When we consider static black holes we use two different coordinates to describe the spacetime, the Schwarzschild coordinates and the Ingoing Eddington–Finkelstein coordinates. Schwarzschild coordinates: Using the Schwarzschild coordinates for a spherically symmetric black hole results in the following formulation for the line element, ds2 = − 1 − 2M r dt2 + 1 − 2M r −1 dr2 + r2 dθ2 + r2 sin2 θdφ2 . (2.16) The constant M denotes the mass of the black hole as measured by an observer at infinity. There are two important facts about this metric; first, this metric is asymptotically flat, that is, in the limit where r → ∞, we recover flat spacetime written in spherical coordinates. The second is that there are two singularities, one located at the origin, at r = 0, and the other is apparent in the metric element grr when r = 2M . If we consider future null infinity, and look at its causal past, any boundaries in the past are known as event horizons. The region inside the event horizon is known as a black hole. For a spherically symmetric spacetime, the event horizon is located at r = 2M . This metric produces the ADM variables for the lapse and shift, α = βi = 1− 2M , r 0. (2.17) (2.18) The singularity at the event horizon is known to be a coordinate singularity, we consider the existence of a new set of coordinates to circumvent this issue. Ingoing Eddington–Finkelstein coordinates: A transformation exists to map the Schwarzschild metric onto coordinates that are referred to as horizon adaptive coordinates. Kruskal coordinates [34] exploit a transformation to remove a 30 2.3. BLACK HOLE SPACETIMES coordinate singularity. Let, u = t − r∗ (2.19) v = t + r∗ (2.20) where r∗ = dr r = r + 2M ln −1 . 1 − 2M/r 2M (2.21) This coordinate transformation sets the singularity to −∞. These coordinates are difficult to use in calculations, and some researchers, e.g. Poisson [102], suggests the use of what are known as ingoing (v, r) or outgoing (u, r) Eddington–Finkelstein coordinates. Using the ingoing coordinates, we cover the regions of space that we are most interested in modeling. In these coordinates, the spherically symmetric metric takes on the form: ds2 = − 1 − 2M r dv 2 + 2dvdr + r2 dθ + r2 sin2 θdφ. (2.22) This may be written using a coordinate transformation derived from the null coordinates [103]: ds2 = − 1 − 2M r dt¯2 + 4M ¯ 2M dtdr + 1 + r r dr2 + r2 dΩ2 , (2.23) where t¯ is not the same as t in the Schwarzschild coordinates. There is a relation between the two time coordinates t¯ = v − r = t + 2M ln r −1 . 2M (2.24) Note that the singularity still exists in the coordinate transform itself. Any relation between the two coordinates must be made outside of the event horizon. Here our lapse and shift take the form r , r + 2M α = (2.25) βr = 2M , r (2.26) βθ = 0, (2.27) βφ = 0. (2.28) 31 2.3. BLACK HOLE SPACETIMES Static Limit Ergoregion Outer Horizon Inner Horizon Ring Singularity Figure 2.2: A Schematic for the Axisymmetric Spacetime. 2.3.3 Axisymmetric Spacetime When the black hole is rotating, we have an axisymmetric spacetime. There are two coordinates commonly used to describe this spacetime, the Boyer–Lindquist coordinates, and the Kerr–Schild coordinates [104, 105, 106, 34]. Boyer–Lindquist coordinates: The Boyer–Lindquist coordinates are a generalization of the Schwarzschild coordinates to allow for rotation. The Kerr spacetime written in these coordinates yields the line element, ds2 = − 1 − 2M r Σ 4aM r sin(θ)2 Σ dtdφ + dr2 + Σdθ2 + ✵ sin(θ)2 dφ2 , Σ ∆ (2.29) Σ = r2 + a2 cos(θ)2 (2.30) ∆ = r2 − 2M r + a2 (2.31) ✵ = dt2 − where we use r 2 + a2 + 2a2 M r sin(θ)2 Σ . (2.32) φ is the angle around the symmetry axis, θ is the polar angle and r is again a radial coordinate. There is one outstanding feature of rotating black holes as opposed to the spherically symmetric black holes: a rotating hole is prone to cause frame dragging, an effect that is most strongly realized 32 2.3. BLACK HOLE SPACETIMES when a particle enters a region known as an ergo-region, as seen in Fig. 2.2. A particle, or light, may enter the ergo-region through a surface called the static limit, defined by rstatic = M + M 2 − a2 cos2 θ. (2.33) Inside the ergo-region the time coordinate, t, transitions from a timelike coordinate to a spacelike coordinate [5]. This surface lies outside the outer horizon and any particle that is within this surface must rotate with the black hole. In the case of the Kerr hole, the event horizon coincides with the static limit at two locations, both on the axis of rotation, θ = 0 and θ = π. In the Kerr spacetime when the grr component of the metric goes to zero we have ∆ = 0, leading to two roots, r+ = M+ M 2 − a2 (2.34) r− = M− M 2 − a2 , (2.35) where the first of these is the outer, or event, horizon. Light cannot escape this horizon just as in the spherically symmetric black hole. The second equation describes the inner horizon, or Cauchy horizon as seen in the diagram 2.2 [107, 5]. Since the second solution is either inside or tangent to the event horizon, it is not discussed further. For more details about the surfaces we refer the reader to Wald [108]. For the horizon to be physical we require that −M < a < M . If a exceeds these limits, we have a naked singularity [109], where the singularity of spacetime is not hidden behind an event horizon, and in this case no black hole exists. When a = 0, the static limit, rstatic , correspond to the event horizon, both the ring singularity and the inner horizon correspond to the centre of the black hole. Thus we are left with only an event horizon, which corresponds to the event horizon for a spherically symmetric black hole. Just as we did for the spherically symmetric black hole in Schwarzschild coordinates, we use a set of horizon adaptive coordinates. Our choice is a generalization of the Eddington–Finkelstein coordinates known as the Kerr–Schild coordinates that remove the coordinate pathologies found at the event horizon. 33 2.3. BLACK HOLE SPACETIMES Kerr–Schild coordinates: For the rotating black hole there is a generalization of the ingoing Eddington–Finkelstein coordinates, known as the Kerr–Schild coordinates. Given the coordinate transformation [5, 95]; dφ = dφBL + (a/∆)dr dt = dtBL + 1 − Zk 1+Y dr, − 1+Y −Z 1−Z (2.36) (2.37) where Y = a2 sin2 θ/Σ, Z = 2M r/Σ, k is a non-negative integer which is used to parametrize the family, and the BL subscript denotes the Boyer–Linquist coordinate. It is common to use k = 1 as it is the simplest member of this family of horizon-penetrating coordinates. This choice of parameter gives us the Kerr–Schild coordinates: ds2 = −(1 − Z)dt2 − 2aZ sin2 θdtdφ + 2Zdtdr + (1 + Z)dr2 −2a(1 + Z) sin2 θdrdφ + ρ2 dθ2 + sin2 θ ρ2 + a2 (1 + Z) sin2 θ dφ2 . 2.3.4 (2.38) Symmetries When looking for symmetries of spacetime, we look for some obvious features, such as a metric that is independent of any particular coordinate. Take for example the Minkowski spacetime in Cartesian coordinates. This metric is independent of all coordinates, which indicates that there are at least four symmetries (three spatial and one temporal) in that spacetime. To find the symmetries in general we look for Killing vectors, ξ, which is a vector field that obeys the Killing equation: ∇µ ξν + ∇ν ξµ = 0. (2.39) This shows symmetries of the system, therein implying a conserved quantity [109]. The Killing equation is derived from Lξ gαβ = 0. (2.40) This is the Lie derivative of the metric along the Killing vector fields. Killing vectors are exploited in Chap. 5. 34 2.4. MAGNETOHYDRODYNAMICS 2.4 Magnetohydrodynamics Magnetohydrodynamics (MHD) couples a fluid to an electromagnetic field. We present each of these, hydrodynamics, and electrodynamics before proceeding to describe the ideal MHD condition and then our MHD system. 2.4.1 Hydrodynamics, A Perfect Fluid Fluid models are used to describe continuum systems by their macroscopic thermodynamic properties, including the internal energy and pressure. For a detailed introduction to relativistic fluids, we refer the reader to Schutz [107], Anile [54], Lichnerowicz [110], or Landau and Lifshitz [111]. A perfect fluid does not have shear stresses, dissipation or heat flux. The stress energy tensor is [4, 109, 107] F Tµν = (ρ + P )uµ uν + P gµν , (2.41) where ρ is the energy density, P is the pressure, uµ are the fluid’s 4-velocity, and gµν is the spacetime metric. Both the density and the pressure are evaluated in the rest frame of the fluid. The energy density includes both the rest mass density ρ0 and the internal energy density ρ0 ǫ, ρ = ρ0 + ǫρ0 , (2.42) where ǫ is the energy density per unit baryon mass, hence a dimensionless value. We also express the fluid enthalpy, ω, ω = ρ + P, (2.43) from which we obtain the specific enthalpy, h, h= P ω =1+ǫ+ . ρ0 ρ0 (2.44) The constituent particles of the fluid are characterized by the number density, n = n(xµ ). If we further assume that the fluid has only one-component, we define the rest mass energy density, ρ0 , by, ρ0 = mB n. (2.45) 35 2.4. MAGNETOHYDRODYNAMICS The fluid 4-current, J µ (xν ), becomes J µ = ρ0 u µ . (2.46) The equations of motion for the fluid are derived from two conservation laws; the conservation of baryon number, ∇µ J µ = 0. (2.47) ∇µ T µν = 0. (2.48) and the stress energy, In particular, we consider projections of the conservation of stress-energy along the fluid’s 4-velocity, and into the plane perpendicular to the 4-velocity. uν ∇µ T µν = 0 (2.49) hλν ∇µ T µν = 0, (2.50) where hλν is the tensor that projects tensor quantities into the plane perpendicular to the fluid’s 4-velocity, hµν = gµν + uµ uν . (2.51) Equations (2.49) and (2.50) are 4 equations with 5 unknowns, ρ0 , ǫ, ui , P , so we need an equation of state, P (ρ0 , ǫ) to close the system of equations. Equation of state For our system we require the equation of state to relate the pressure, P , the baryon rest mass density, ρ0 , and the energy density ǫ. P → P (ρo , ǫ). From the work by Chandrasekhar [112] and Synge [113], we have an ideal gas equation of state for a relativistic non-degenerate adiabatic monatomic gas, P (ρ0 , ǫ) = (Γ − 1)ρ0 ǫ, (2.52) 36 2.4. MAGNETOHYDRODYNAMICS where Γ is the adiabatic constant. The cases of interest include Γ = 5/3 for a low energy nonrelativistic gas, and Γ = 4/3 for a high energy relativistic gas [113, 112]. These are limiting cases for the more general equation of state described by Taub [114], where the adiabatic constant depends on the baryon rest mass density. Taub [114] also shows that the ideal gas equation of state is only physically valid below the limit Γ ≤ 5/3 if the system is to remain in agreement with the relativistic kinetic theory of gases [115]. For ultrarelativistic systems, where we take the limit, ρ = ρ0 (1 + ǫ) → ρ0 ǫ. (2.53) The ideal relativistic equation of state is defined as P = P (ρ) = (Γ − 1)ρ, (2.54) where ρ is the internal energy density of the fluid. This is a barotropic equation of state, since the pressure is a function of only the internal energy density, and the use of this equation of state allows us to determine the equations of motion of a hydrodynamic system by using Eq. (2.48) alone. This is possible because we are assuming the rest mass density conservation, Eq. (2.47), is dynamically unimportant. We use the ultrarelativistic equation of state, Eqn. (2.54) when the internal energy of the fluid is much greater than the fluid’s rest mass density, such as expected for a radiation fluid. The enthalpy for the ultrarelativistic system may be written as ω= ΓP . Γ−1 (2.55) The Speed of Sound By studying a linearized form of the hydrodynamic equations of motion, we can determine a measure of how fast a disturbance travels in a fluid medium. This speed is referred to as the speed of sound; specifically, given the equation of state, we can determine the speed of sound, cs , by taking the partial derivative with respect to the internal energy density [111], c2s = ∂P ∂ρ . (2.56) s 37 2.4. MAGNETOHYDRODYNAMICS Where ρ is the fluid energy density, and the s on the right hand side denotes the entropy per particle. When applying Eqn. (2.56) to the ideal gas equation of state in the form (2.52) we get; c2s = ΓP . ρ0 h (2.57) In the Newtonian limit ρ0 h → ρ, we recover the usual speed of sound in an ideal gas, c2s = ΓP/ρ. For the ultra relativistic fluid equation of state (2.54) we have; c2s = Γ − 1. (2.58) In our ultrarelativistic study we restrict our attention to the adiabatic constant Γ = 4/3. When we study magnetohydrodynamics we will use both Γ = 4/3 and Γ = 5/3. This is the extent of the information that we need from hydrodynamics alone. We now proceed by describing the electromagnetic properties needed to describe magnetohydrodynamics. 2.4.2 Electromagnetism Since we are considering ideal MHD in this research, we begin by describing the electromagnetic stress-energy tensor, and then describe the relativistic ideal MHD condition, then finally the full ideal MHD stress-energy tensor. For an electromagnetic field in vacuum, we have the well-known electromagnetic stress-energy EM tensor Tµν [4, 109, 107, 116]; 1 EM Tµν = Fµσ Fν σ − gµν Fαβ F αβ , 4 (2.59) where Fµν is the Faraday tensor. Adopting a locally inertial frame in Cartesian coordinates we have, Fµν 0 Ex = Ey Ez −Ex −Ey 0 Bz −Bz 0 By Bx −Ez −By . Bx 0 (2.60) Ei are the electric field components, and Bi are the magnetic field components. The mixed form of the Faraday tensor is defined as F σ ν = g σµ Fµν . Just as with the hydrodynamic stress-energy 38 2.4. MAGNETOHYDRODYNAMICS tensor (2.41), it can be shown that in a local inertial frame, the element T00 = (E 2 + B 2 ) is the local energy density made up of the electric field energy density E 2 , and the magnetic field energy density B 2 . The elements T0j denote the contributions to the stress-energy from the Poynting flux, the momentum associated with an electromagnetic field. The elements Tij are the elements of the usual non-relativistic Maxwell stress tensor. It is convenient to define the Maxwell tensor ∗Fµν = ǫµνλκ F µν , where ǫµνλκ = √ −g[µνλκ], and where [µνλκ] is the Levi–Civita totally antisymmetric tensor, with components (±1, 0), and F µν = g µγ g νω Fγω . The relativistic electrodynamic equations of motion are completely described by [116], ∇µ F µν = J µ , (2.61) ∇µ ∗F µν = 0, (2.62) where (2.61) is the generalization of the Maxwell equations that govern the dynamics and constraints of the electric field, and (2.62) is the generalization the Maxwell equations that govern the dynamics and constraints of the magnetic field [116]. J µ is the electromagnetic 4-current. 2.4.3 Relativistic Force Free Condition In ideal MHD, we consider the magnetic field as frozen to the fluid, such that the number of field lines passing through a fluid element is constant. For this behaviour to occur, the electric field in the co-moving frame of the fluid must be zero. This is known as the force free condition, and may be derived by setting the Lorentz force in the rest frame to zero, Fµν uν = 0, (2.63) where we have projected the electromagnetic field into the frame of the fluid. The assumption of frozen flux lines is valid for many astrophysical phenomena since the conductivity in these systems is extremely large. The ideal MHD condition assumes that the electric field is dependent on the velocity and the magnetic field, thus allowing us to eliminate the electric field from the Faraday tensor, and eliminating the need to evolve the electric fields via Eqns. (2.61). This is only true in the ideal MHD limit, the case of infinite conductivity. 39 2.4. MAGNETOHYDRODYNAMICS The electromagnetic stress-energy tensor (2.59) may now be expressed in terms of the fluid 4-velocity, uλ , and the magnetic fields alone. In the reference frame of the fluid, we have the electromagnetic stress energy contributions T00 = B 2 , Tjk = Bj Bk + (B 2 /2)δjk , the remaining terms being zero. When formulating the stress energy tensor for ideal MHD, it is convenient to define the magnetic 4-vector, bµ , by projecting the Faraday tensor into the direction of the fluid 4-velocity [55, 54]: bµ = uν ∗F µν , (2.64) where the index on the 4-vector bµ is raised and lowered using the 4-metric bµ = gµν bν . The components of the 4-vector are found seen in (2.99). With this definition we may write the electromagnetic stress-energy tensor (2.59) EM Tµν = b2 u µ u ν + b2 gµν − bµ bν , 2 (2.65) in analogy to the hydrodynamic stress energy tensor we consider b2 = gµν bµ bν to be the magnetic energy, and b2 /2 to be the magnetic pressure. We are now in a position to determine the stress-energy tensor for ideal MHD by adding the two tensors together; 1 MHD F EM Tµν = Tµν + Tµν = (ρ + P + b2 )uµ uν + (P + b2 )gµν − bµ bν . 2 (2.66) Since we assume that the original hydrodynamic model applies to a neutral fluid, any interaction terms between the fluid and the magnetic field are further assumed to be negligible. We carry on the analogy between the hydrodynamic stress-energy tensor and the MHD stressenergy tensor and define the MHD enthalpy, h∗ , h∗ = h + b2 , ρ0 (2.67) P∗ = P + b2 . 2 (2.68) as well as the MHD pressure, P ∗ , For the remainder of this thesis, when we refer to a stress-energy tensor, we will mean the MHD stress energy tensor (2.66). 40 2.5. DERIVATION OF THE EQUATIONS OF MOTION Since the local electric field is zero in the ideal MHD approximation, it suffices to include Eqns. (2.62) in the development of the equations of motion for and ideal MHD system. The coupling between magnetic fields and fluid matter leads to the production of other types of disturbances, known as Alfv´en waves and magnetosonic waves. We briefly discuss these here. The Alfv´ en Speed By studying a linearization of the MHD equations of motion, Alfv´en [117] proved the existence of a wave that arises from the coupling of matter and a magnetic field. This wave is a transverse, non-compressive wave that balances the restoring force of tension in locally curved magnetic field lines against the mass inertia of the fluid trapped in the field. It is referred to as the Alfv´en wave or shear Alfv´en wave and travels at a speed [54], ca , c2a = |b|2 , ρ0 h ∗ (2.69) along the magnetic field lines. When we consider the Newtonian limit, we recover, |b|2 → B 2 , and ρ0 h∗ → ρ, so the Alfv´en speed reduces to c2a = B 2 /ρ, which is the Newtonian expression of the Alfv´en speed [97]. The restoring force for the Alfv´en wave is the magnetic pressure, and the velocity as defined in Eqn. (2.69) is directly analogous to the speed of sound as presented in Eqn. (2.57). Magnetosonic Speeds Other waves that arises as a result of the coupling of matter and a magnetic field are the magnetosonic or magneto-acoustic waves. These waves are longitudinal, compressive waves that balance the restoring force of both the the magnetic and fluid pressure against the mass inertia of the fluid. The speeds of the magnetosonic waves, cms , are the roots of a quartic function defined in Chap. 3 Eqn. (3.49). Both the magnetic field and the pressure play a role in defining these waves speeds. The four roots to the quartic function are classified as fast, cms, f , and slow, cms, s so named because the fast wave has a larger magnitude than the Alfv´en speed and, the other, a smaller magnitude. 2.5 Derivation of The Equations of Motion The fluid equations (2.49) are not readily amenable to the Cauchy problem, so we need to cast them into a form that may be solved numerically. We now derive the conservative form of the 41 2.5. DERIVATION OF THE EQUATIONS OF MOTION equations of motion, using the ADM line element, which, when considering a coordinate system xµ as defined earlier, may be written: ds2 = −(α2 − β i βi )dt2 + 2βi dtdxi + γij dxi dxj . (2.70) We follow the derivation of Font [21]. Once the spacetime has been sliced we use the definition of the lapse, α, and shift β i , to describe the fluid motion relative to stationary observers on the spacelike hypersurfaces, Σt . The observers in this frame are referred to as Eulerian observers [36]. Observers at rest in the hypersurface, Σt , have 4-velocity nµ . They will measure the fluid velocity as, ui . −nν ui (2.71) ui ui βi = + , µ −n uµ W α (2.72) vi = The contravariant components, v i = γ ij vj , are vi = where ui are the spatial components of the fluid 4-velocity, uµ . The Lorentz factor, W , is given by the relation W = −nµ uµ = αut which may also be expressed as; W = 1 . 1 − γij ui uj (2.73) We define our 3 + 1 conservative variables, density, D, energy, E, momentum, Si , and magnetic field, B µ as: D = −nν J ν ; (2.74) E = T µν nµ nν ; (2.75) Si = T µν nµ γνi ; (2.76) Bµ = −nν ∗F µν . (2.77) The variables q are locally conserved quantities, allowing the equations of motion to be written in conservative form, which is essential for the numerical methods described in Chap. 3. Explicit 42 2.5. DERIVATION OF THE EQUATIONS OF MOTION forms for the conservative variables, when considering our normal 4-vector (2.12) are: D = ρ0 W ; (2.78) E 1 = (ρ0 h + b2 )W 2 − (P + b2 ) − (αbt )2 ; 2 (2.79) Si = (ρ0 h + b2 )W 2 vi − αbt bi ; (2.80) Bµ = α ∗F tν . (2.81) It is typical, when studying the relativistic hydrodynamic equations, to define a new variable, τ, τ ≡ E − D. (2.82) This variable, when taken to the Newtonian limit, is the classical energy density. The first fluid equation we calculate is the conservation of baryon density (2.47), ∇µ J µ = 0 ⇒ ∇µ (ρ0 uµ ) = 0, (2.83) which may be reformulated as; (ρ0 uµ ),µ +(ρ0 uα )Γµ αµ = 0. Γµ αν are the Christoffel symbols which is determined by [4] Γµ αν = 1 µδ g (gδα,ν + gδν,α − gαν,δ ) . 2 (2.84) There is a well known relation for a Christoffel symbol contracted over the upper and one of the lower indices; Γ µ αµ √ ( −g),α . = √ −g (2.85) Using (2.74) and (2.85) we arrive at √ √ 1 √ −gρ0 ut ),t +( −gρ0 ui ),i = 0. −g (2.86) 43 2.5. DERIVATION OF THE EQUATIONS OF MOTION Then using (2.78) and (2.72), we obtain the final form for the conservation of baryon density: √ √ βi ∂t ( γD) + ∂i −gD v i − α 1 √ −g = 0. (2.87) We will now explicitly project the equations of conservation of stress-energy into the 3+1 hypersurfaces as described in Sec. 2.2, ∇µ T µν = 0. (2.88) A typical development of the 3 + 1 fluid equations of motion requires the projection of (2.88) into the plane perpendicular to the spacelike hypersurfaces, Σt , to give the equation of motion for the energy, and in the plane parallel to the hypersurfaces to give the Euler equations: γjν ∇µ T µν = 0; (2.89) nν ∇µ T µν = 0. (2.90) We write 2.89, 2.90 in conservative form: ∇µ (T µν γjν ) = (∇µ T µν )γjν + T µν ∇µ γjν , (2.91) ∇µ (T µν nν ) = (∇µ T µν )nν + T µν ∇µ nν , (2.92) where the first term on the right of Eqns. (2.91) and (2.92) is zero by conservation of the stress energy tensor (2.88). We will also make use of the two expressions: ∇µ γjν = ∂µ γjν − Γδ µν γjδ − Γδ jµ γδν ; (2.93) ∇µ nν = ∂µ nν − Γδ µν nδ . (2.94) Using (2.93), (2.94) and (2.85) we get ∂µ √ ∂µ √ −gTνµ nν −gTiµ − 1√ −gT µν ∂i gµν = 0; 2 √ − −gT νµ ∂µ nν = 0. (2.95) (2.96) 44 2.5. DERIVATION OF THE EQUATIONS OF MOTION Equations (2.95) and (2.96) reduce to √ √ 1 √ ( −gSj ),t +( −g(Sj vˆi + P δ i j − bj B i /W )),i = T µν (giν,µ − Γδ νµ giδ ), −g √ √ 1 √ ( −gτ ),t +( −g(τ vˆi + P v i − αbt B i /W )),i = −T µν Γt νµ . −g (2.97) (2.98) Here, for notational simplicity, we have defined vˆi = v i − β i /α. The terms bj = gjν bν , and bk and bt are the spatial and temporal components of the previously defined magnetic 4-vector, which may be explicitly written as Bj W (γik v i B k ), + αbt vˆj . α W bµ = (bt , bj ) = (2.99) Eqns. (2.97) and (2.98) are the general flux conservative form for the momentum and energy, respectively. To determine the evolution and constraint equation for the magnetic field, we project (2.62) into a component in the hypersurface and the component perpendicular to the hypersurface: nν ∇µ ∗F µν = 0; (2.100) ⊥i ν ∇µ ∗F µν = 0, (2.101) From (2.100) we get the no-magnetic-monopole constraint ∂i √ γB i = 0, (2.102) and from (2.101) the magnetic induction equation ∂t √ √ i γB − ∂j −g(B i vˆj − B j vˆi ) = 0. (2.103) 45 2.6. CONSERVATION OF THE DIVERGENCE FREE MAGNETIC FIELD We define the complete set of conserved quantities: D Sj q= τ Bk (2.104) We have the corresponding flux terms: Dˆ vi Sj vˆi + P δ i − bj B i /W j i f (q) = τ vˆi + P v i − αbt B i /W B k vˆi − vˆk B i , (2.105) where vˆi = v i − β i /α. We express the source terms on the right hand side of (2.87), (2.97), (2.98), and (2.103) as a source vector: 0 T µν gjν,µ − Γδ gjδ νµ S(q; gµν ) = µt µν t T α,µ − T Γνµ 0 . (2.106) Using the above vectors we write the generalized conservative equation ∂ √ ∂ √ 1 1 √ γq+ √ −g f i (q) = S(q; gµν ), 0 −g ∂x −g ∂xi (2.107) which has the form necessary for the use of the high resolution shock capturing methods that will be described in Chap. 3. 2.6 Conservation of the Divergence Free Magnetic Field Since the evolution of the magnetic field has a constraint ∇ · B = 0, we have three evolution equations and one constraint equation. This leaves us with an over determined set of equations. Traditionally, the numerical treatment of a constrained system uses free evolution, where the evo- 46 2.6. CONSERVATION OF THE DIVERGENCE FREE MAGNETIC FIELD lution equations are used to evolve the system of equations and there is an implicit assumption that the constraint will be maintained. However, when using free evolution, any numerical errors that arise are often linked to constraint violations, thus we need a method that enforces the constraint as the flow evolves. Conservation of the ∇ · B = 0 constraint is difficult to apply when studying numerical MHD in general. For MHD evolution in the Newtonian limit, Hawley and Stone [66] have a comprehensive list of the techniques and their advantages and disadvantages. The methods used for general relativistic treatments are the same as those in the Newtonian systems, just written in generalized coordinates. Deviations from the divergence condition arise from the discretization approximation and once produced may lead to unphysical wave transport properties such as transport of Alfv´en waves orthogonal to the magnetic field [118]. If we were to evolve the system with no consideration paid to the constraint violations, we would need to ensure that the violations remain bounded and that the error in the divergence is comparable to the truncation error in the numerical approximation. Since we cannot easily control the bounds on the error we focus our attention on techniques that will conserve the constraint during the evolution. The technique we implement to achieve this goal was the diffusive hyperbolic divergence cleaning.11 We describe the diffusive hyperbolic divergence cleaning method, where the ∇ · B = 0 violations are transported out of the domain with a finite speed. 2.6.1 Divergence Cleaning There are three techniques that are classified as divergence cleaning methods: 1. elliptic divergence cleaning [121, 122]; 2. parabolic divergence cleaning [118, 122]; and 3. diffusive hyperbolic divergence cleaning [118, 122]. We focus on the diffusive hyperbolic divergence cleaning method, and outline the technique below. 11 We are aware that other methods such as constrained transport [119], and flux-CT [120] exist, but chose to try a method that easily allows for the future use of adaptive mesh refinement. The other methods mentioned may be used with AMR; however, they are more complicated. 47 2.6. CONSERVATION OF THE DIVERGENCE FREE MAGNETIC FIELD Diffusive Hyperbolic Divergence Cleaning Diffusive hyperbolic divergence cleaning was investigated in non-relativistic MHD by Dedner [118] and later in special relativistic MHD by Anderson et al. [122]. The diffusive-hyperbolic technique has the advantage that it is local, so it does not require averaging over a wider range of cells, which is a requirement of other constraint maintenance techniques such as the so-called “flux-CT” method [120]. Another advantage is that it adheres to the hyperbolic form, and consequently the finite volume method. The last advantage is that it is easy to implement numerically [122, 123]. To implement the diffusive hyperbolic method, we add a divergence term acting on an auxiliary field ψ of the form ∇µ (g µν ψ) to the Maxwell equations, ∇µ (∗ F µν + g µν ψ) = 0. (2.108) We also add a diffusive term, κnµ ψ to (2.108) to damp out any ∇ · B = 0 violations [124], where κ is a tunable parameter: ∇µ ( ∗F µν + g µν ψ) = −κnν ψ. (2.109) To understand why Eqn. (2.109) removes ∇ · B = 0 violations we take the divergence of Eqn. (2.109) [124], ∇ν ∇µ ( ∗F µν + g µν ψ) = −κ∇ν (nν ψ), (2.110) and use the fact that the Faraday tensor is anti-symmetric we get, ∇2 (ψ) = −κ∇ν (nν ψ). (2.111) To emphasize our point, when we consider flat spacetime, Eqn. (2.111), becomes ∂ν ∂ ν ψ = −κ∂ν (nν ψ). (2.112) Equation (2.112) has the form of a damped wave equation for ψ travelling at the speed of light, provided that κ > 0 [124]. The wave equation damps ψ to zero, which recovers the original induction equation and no-magnetic-monopole constraint. 48 2.6. CONSERVATION OF THE DIVERGENCE FREE MAGNETIC FIELD Projecting equation (2.108) into 3 + 1 form, we get ∂t √ √ γ Πj + ∂i ( −g (v i B j − v j B i + αg ij Ψ)) Bi βi − Ψ α α √ √ −g ∂t ( γ Ψ) + ∂i √ = −α −gg αµ Γjαµ Ψ + κβ j Ψ, (2.113) √ = α −gg αµ Γtαµ Ψ − κΨ, (2.114) where we have defined two new fields, Πi ≡ Ψ ≡ B i + Ψβ i , (2.115) ψ/α (2.116) to maintain the structure of the conservative expressions. The waves from this new field, Ψ, travel at the speed of light and dissipate at some rate κ, which is a tunable parameter. Some researchers also leave the propagation speed of Ψ as a tunable parameter; however, in our system, we found that propagation speed of the violation must be as fast as possible to keep the solution physical, and to transport violations of the constraint out of the domain as fast as possible. Since the diffusive hyperbolic method augments the original set of conservation equations of motion, we restate the new state vectors, D Sj q= τ k Π Ψ , (2.117) and their associated flux, f i and source vectors, s: Dˆ vi Sj vˆi + P δji − bj B i /W i i t i f (q) = τ vˆ + P v − αb B /W k i B vˆ − vˆk B i + αg ij Ψ B i /α − Ψβ i /α , (2.118) 49 2.7. ULTRARELATIVISTIC EQUATIONS OF MOTION 0 √ −gT µν giν,µ − Γδνµ giδ √ −g T µt α,µ − T µν Γtνµ S(q; gµν ) = −αg αµ Γj αµ Ψ + κβ j Ψ αg αµ Γt αµ Ψ − κΨ , (2.119) where vˆi ≡ v i − β i /α. Now our state vector has the form ρ0 j v p= P k B Ψ . (2.120) Equations (2.117), (2.118), and (2.119) together with (2.107) form our equations of motion for the general relativistic magnetohydrodynamic system. 2.7 Ultrarelativistic Equations of Motion In the introduction we stated that we would be studying two distinct fluid models, one modelled using ultrarelativistic hydrodynamics, the other modelled using magnetohydrodynamics. We have seen the magnetohydrodynamic equations of motion described above. Now we turn our attention to the ultrarelativistic hydrodynamic description where we are not concerned with fluids with an embedded magnetic field. When we describe the velocity of a fluid element, we are describing the net velocity of the constituent particles of the fluid cell. For the fluids described so far there have not been any assumptions on the motion of the individual particles. However, when we describe an ultrarelativistic fluid we are describing a fluid made up of elements in which the constituent particles are individually moving at relativistic velocities. In ultrarelativistic systems the thermal energy of the fluid 50 2.7. ULTRARELATIVISTIC EQUATIONS OF MOTION dominates the rest mass energy. We take the limit ρ = ρ0 + ρ0 ǫ → ρ0 ǫ, ρ0 h = ρ0 + ρ0 ǫ + P → ρ0 ǫ + P = ρ + P, (2.121) where ρ is the internal energy of the fluid [125, 123]. The procedure to derive the equations of motion for the ultrarelativistic system is almost identical to the previously outlined procedure for the magnetohydrodynamic equations of motion. Since we take the limit in which we may neglect the rest mass density we neglect the continuity, Eqn. (2.87). For simplicity we also neglect the magnetic field contributions. The conservative variables are: 2 Si (ρ + P )W vi q= . = 2 (ρ + P )W − P τ The flux vectors are: The source vector elements are: i P δji Sj vˆ + f i (q) = τ vˆi + P v i (2.122) . √ µν giν,µ − Γδνµ giδ −gT S(q; gµν ) = √ −g T µt α,µ − T µν Γtνµ (2.123) , (2.124) with the equation of state P = (Γ − 1)ρ. The equations of motion remain of the form: 1 1 ∂ √ ∂ √ √ γq+ √ −g f i (q) = S(q; gµν ). 0 −g ∂x −g ∂xi (2.125) Equations (2.122), (2.123), (2.124), together with (2.125), form the equations of motion for our ultrarelativistic hydrodynamic system. Dimensionless Parameters Here we describe the dimensionless control parameters that are used to parametrize the system under study. These are measurable parameters which will be used to characterize the system we are modeling. There are two parameters that are used to specify our hydrodynamic system, these 51 2.7. ULTRARELATIVISTIC EQUATIONS OF MOTION are: • asymptotic Mach Number (M∞ ); • Adiabatic Constant (Γ). The Mach number is the ratio of the velocity of the fluid and the speed of sound in the fluid. v . cs M= (2.126) The asymptotic Mach number is then the ratio of the asymptotic velocity of the fluid and the asymptotic speed of sound, M∞ = v∞ . c∞ s (2.127) When considering the relativistic systems, the Mach number is defined as [126] MR = Wv W = M, Wc cs Wc (2.128) and we define the asymptotic relativistic Mach number as, MR ∞ = where Wc = 1 − c2s W = 1−v 1 2 −2 − 12 W∞ W∞ v∞ = M∞ , Wc∞ c∞ W c∞ s (2.129) is the sound speed equivalent of the relativistic Lorentz factor [1], and is the relativistic Lorentz factor. The relativistic Mach number is the ratio of the proper speed of the fluid with the proper speed of sound in the fluid. When considering magnetic field contributions, we have one more parameter that may be used to specify the system: • Plasma beta parameter (βP ) βP = 2P , |b|2 (2.130) is the ratio of the hydrodynamic pressure to the magnetic pressure. For the systems we study, this value is determined by the asymptotic values of P , and |b|2 , denoted P∞ and |b∞ |2 respectively. In Chap. 5 we use βP∞ = 2P∞ /|b∞ |2 , a tunable quantity and will be used to parametrize the magnetic fields as we include them in the MHD effects. For typical MHD systems, we have that βP is on the order of unity. For weakly magnetized systems βP → ∞, and the MHD treatment is no 52 2.8. GEOMETRIC CONFIGURATIONS longer physically necessary as the (electro)magnetic fields would act as test fields on a geometric background. In systems where βP ≪ 1, the ideal MHD approximation is no longer self consistent, and the coupled fluid-electromagnetic system should be described by force free electrodynamics [124, 127].12 2.8 Geometric Configurations In this thesis, we consider an axisymmetric spacetime. The computational costs in equipment and time would be too great to do a full 3+1 calculation, so we impose either an axisymmetry, or an infinitely thin-disk approximation on our system. All problems considered here are 3 + 1 type systems, but, where by virtue of a coordinate symmetry the geometry may be completely specified by two free spatial coordinates, we will refer to them as 2D systems where the time dependence is implicit. When considering the motion of a black hole through a homogeneous background fluid, we need to consider the possibility of the fluid having an angular velocity relative to the motion of the black hole, thus this type of motion requires a minimum of two spatial dimensions. We investigate two different geometries, where the fluid may have an angular velocity relative to the black hole, an axisymmetric flow, and what is known as the infinitely thin-disk approximation. In previous studies [1, 2], there were problems, such as coordinate singularities, in studying accretion near the event horizon. When so-called horizonpenetrating coordinates are used, the coordinate singularities were alleviated. For this study of the Bondi–Hoyle problem, we only use horizon-penetrating coordinates. In the Bondi–Hoyle problem the background fluid is at rest and it is the accretor that is moving through the fluid. We use that setup to describe the different symmetries used in this study. Axisymmetry When the flow is considered to be symmetric around the axis defined by the trajectory of the black hole we refer to this as axisymmetric flow. This is in contrast to a stationary spherically symmetric body in a uniform fluid background, which is completely specified by one spatial dimension, the radius r. Axisymmetry is used for both a spherically symmetric black hole travelling through the fluid background as well as an axisymmetric black hole with the rotation axis aligned with 12 Force-free electrodynamics assumes that the matter contributions to the fluid dynamics are no longer important and that the fluid is no longer a perfect conductor. The motion of the system is determined by the full set of Maxwell equations, as well as the momentum equations in the limit that the fluid density is constant. 53 2.8. GEOMETRIC CONFIGURATIONS the direction of the trajectory. To describe spacetime in axisymmetry, we use the Kerr–Schild coordinates, (2.38), where we restrict the azimuthal coordinate to φ = const. Infinitely Thin–Disk Approximation The infinitely thin–disk approximation is used to restrict the evolution of the flow to the equatorial plane. We assume this geometry for simplicity [2, 3]. We assume that the structure of the flow does not depend on the polar coordinate and that the polar pressure gradients and polar velocities are negligible. In non-relativistic models this geometry is considered as wind accretion past an infinite cylinder using cylindrical coordinates [90, 128]13 . We use the Kerr–Schild coordinates to describe this geometry, where in this case we will restrict the polar coordinate to θ = π/2, thereby restricting the flow to the equatorial plane of the black hole. Since we restrict the polar coordinate, the polar vector elements are constant and the flux out of the plane is zero. The axis of rotation of the rotating black hole is orthogonal to the direction of the fluid flowing from infinity. The infinitely thin-disk approximation is highly restrictive and is used to gain a basic understanding of the flow morphology. To obtain a full understanding of the nature of accretion onto a rotating black hole, a full three dimensional evolution will be necessary. 13 As cited by Font et al. [3] 54 CHAPTER 3 FINITE VOLUME METHODS 3.1 Introduction In this chapter, we describe the numerical methods used to solve two sets of equations, the ideal magnetohydrodynamic (MHD) equations of motion discussed in Sec. 2.5, and the ultrarelativistic (UHD) equations of motion discussed in Sec. 2.7. We define hyperbolic partial differential equations and the characteristics of our systems of equations. We discuss weak solutions to hyperbolic systems, before describing the particulars of the numerical approximations used to solve the hydrodynamic equations of motion, such as Godunov’s method and generalizations of Godunov’s method used in the finite volume schemes. In general, the equations for non-dissipative compressible flows will develop discontinuities, which arise from discontinuous changes in the fluid pressure. Since the pressure gradients in such situations are infinite, we require a specially designed numerical technique that would allow for discontinuities in the fluid variables q or p, as described in the beginning of Chap. 2. If discontinuities appear in the system, centred finite difference schemes fail, since derivatives across discontinuities are ill-defined. Godunov [23] developed a method to handle the discontinuities, by dividing the domain of integration into small cells and treating each cell as a Riemann problem, (defined in 3.6). Godunov’s method was later extended to higher accuracy schemes, known as high-resolution shock-capturing methods based on th finite volume method (FVM). Finite volume methods are a technique used to discretize hyperbolic partial differential equations. By using specialized flux calculations, FVMs may be extended to high-resolution shockcapturing schemes (HRSC), and in turn may be used as a generalization of the Godunov approach for solving hyperbolic systems of equations over discontinuous initial data. The HRSCs follows Godunov’s prescription by breaking the domain of integration into finite volumes or cells. The finite volume method calculates an integral solution to the original PDE problem over each of these cells, the basics of which we present below. This is by no means expected to be a definitive reference for 55 3.2. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS this technique, as there are several good books on this topic. For more details on the finite volume method we refer the reader to LeVeque [19]. 3.2 Hyperbolic Partial Differential Equations The finite volume method requires that the hyperbolic equations be written in a general conservation form for the solution vector, q = q(xµ ), ∂t q + ∂i f i (q) = s(q), (3.1) where f i (q) = f i (q(xµ )) is the flux density vector, and s = s(q(xµ ), xµ ) is the source vector. Since the source terms do not have any bearing on the special techniques used in the finite volume method we will drop the source term in the next several sections for convenience. Hyperbolicity Expanding the divergence in equations of the type (3.1), we get ∂t q + ∂f i ∂i q = 0. ∂q (3.2) This form is known as the quasilinear form of the conservation equations, due to the fact that the PDE’s are linear in their first derivative; however, the Jacobian ∂f i /∂q is dependent on the variables q and is itself nonlinear in our system. We re-write the Jacobian as a matrix Ai = Ai (q), ∂f i = Ai . ∂q (3.3) Thus we have one Jacobian matrix per spatial dimension. Equation (3.2) becomes ∂t q + Ai ∂i q = 0. (3.4) The system of equations (3.4) is defined to be hyperbolic if all the eigenvalues of each of the Jacobians are real. The nature of the eigenvalues of each of the Jacobians leads to the definition of several notions of hyperbolicity [19, 129]. These notions apply to the Jacobian of each dimension individually: 56 3.3. CALCULATING THE PRIMITIVE VARIABLES • Strongly hyperbolic – the Jacobian is diagonalizable; • Symmetric hyperbolic – the Jacobian is symmetric and diagonalizable; • Strictly hyperbolic – the eigenvalues of the Jacobian are all distinct, and the Jacobian is diagonalizable; • Weakly hyperbolic – the Jacobian has degenerate eigenvalues, and the Jacobian is diagonalizable. By diagonalizable we mean that each of the Jacobians, Ai , can be written in the form A = RΛR−1 , (3.5) where Λ is a diagonal matrix, and R is the set of right eigenvectors. If the Jacobian is strongly hyperbolic, it is known to be well-posed. Well-posed problems are mathematical equations that are known to have a solution that is unique and depends continuously on the data. As cited by Putten [130], Friedrichs [131] proves that the ideal hydrodynamic equations of motion for special relativistic systems are well-posed. We assume this will also hold true for general relativistic systems. The advantage of studying hyperbolic partial differential equations is that the solutions are expected to have wavelike behaviour. The solutions to these systems depend on different wave speeds, which correspond to the eigenvalues, or characteristics, of the Jacobian of the system of equations. In Sec. 3.4 we describe the spectral decomposition of the Jacobian for the full system of general relativistic magnetohydrodynamic (GRMHD) equations. The calculation of the characteristics in Sec. 3.4 requires that we know both the conservative variables, q = q(p), and primitive variables, p = p(q), as defined in 2.1. In the next section we outline the procedure used to perform the variable transformation p = p(q). 3.3 Calculating the Primitive Variables When performing magnetohydrodynamic calculations we typically record the equations of motion in terms of the conservation laws of energy-momentum, Eqn. (2.48), and matter, Eqn. (2.47). Written in this form the variables describing the fluid take on a particular form. We call this form of the variables “conservative”. However, when describing fluid flow we rarely directly discuss the 57 3.3. CALCULATING THE PRIMITIVE VARIABLES energy and momentum of the fluid elements, but rather the pressure, velocities, etc. We refer to the latter variables as “primitive”. In relativistic magnetohydrodynamics we are often required to use both variables in our calculations. Due to the highly nonlinear nature of this fluid model we are as yet unable to find a closed form solution for the variable conversion and thus rely on numerical techniques. We used the method derived by del Zanna, as outlined below [125]. The process of calculating the primitive variables from the conservative variables is known as primitive variable recovery14. We define Ω ≡ ρ0 hW 2 , (3.6) where ρ0 is the baryon rest mass density, h is the fluid enthalpy, and W is the relativistic Lorentz factor. We also define Q ≡ Si B i , (3.7) the scalar product of the fluid’s 3-velocity and the magnetic field. The first stage of the primitive variable recovery is to express the conservative variables as functions of Ω, as outlined below. We write the energy equation (2.82) as τ = E−D = Ω−P + 1− 1 2W 2 |B|2 − Q2 − D, 2Ω2 (3.8) and using the momentum equation (2.80) we get |S|2 (Ω) = γ ij Si Sj = (Ω + |B|2 )2 1 − 1 W2 − Q2 (2Ω + |B|2 ). Ω2 (3.9) We solve (3.9) for W W (Ω) = 1 − Q2 (2Ω + |B|2 ) + |S|2 Ω (Ω + |B|2 )2 Ω2 − 12 , (3.10) then re-formulate the equation of state, P = (Γ − 1)ǫρ0 as P (Ω) = (Ω − DW )(Γ − 1) . ΓW 2 (3.11) 14 We are aware of other techniques to solve for the primitive variables such as a two variable and five variable solver also outlined in Noble et al. [125]; however, after testing, the single variable solver was found to be simpler and just as robust for the systems we investigated. 58 3.3. CALCULATING THE PRIMITIVE VARIABLES Using (3.8), we define f (Ω) as f (Ω) ≡ Ω − P + 1 − 1 2W 2 |B|2 − Q2 − D − τ = 0. 2Ω2 (3.12) Finally, we use Newton’s method to iteratively solve the nonlinear (3.12) for Ω, ΩI+1 = ΩI − f (ΩI ) , f ′ (ΩI ) (3.13) where I denotes the Newton iteration, and prime denotes derivatives with respect to Ω f ′ (ΩI ) = ∂f I (Ω ). ∂Ω (3.14) To calculate ∂f /∂Ω, we use the relations listed below: df dΩ dP dΩ dW dΩ |B|2 dW Q2 ; + W 3 dΩ Ω2 (W (1 + DW ′ ) − 2ΩW ′ ) (Γ − 1) ; ΓW 3 = 1 − P′ + (3.15) = (3.16) = −W 3 2Q2 3Ω2 + 3Ω|B|2 + |B|4 + |S|2 Ω3 3 2Ω3 (Ω + |B|2 ) . (3.17) We perform this procedure at every time step, n. The Newton iteration must be initialized, so as an initial guess for the value of Ω0 we used the definition of Ω (3.6) and values for ρ0 , h, and W from the previous timestep, n − 1: Ω0 = ρ0n−1 hn−1 /(1 − γij vin−1 vjn−1 )2 . (3.18) Using this initial guess we were able to increase the speed at which the solver converged to the correct solution. Occasionally this method may fail to converge to a solution. In such situations another method was used to determine the primitive variables; this is discussed in Sec. 3.9.1. Now that we have Ω, we solve for the primitive variables. Using (2.80) we express the velocity components as vi = 1 Ω + |B|2 Si + Q Bi , Ω (3.19) 59 3.4. CHARACTERISTICS and the Lorentz factor W (3.10) as W = 1− Q2 (2Ω + |B|2 ) + |S|2 Ω (Ω + |B|2 )2 Ω2 − 21 . We use the equation of state as expressed above (3.11) to calculate the pressure: P (Ω) = (Ω − DW )(Γ − 1) . ΓW 2 We recover the baryon rest mass density, ρ0 = D . W (3.20) When using diffusive hyperbolic divergence cleaning, as defined in Sec. 2.6.1, we recover the magnetic field density B i = Πi − Ψβ i . (3.21) Ultrarelativistic Primitive Variable Recovery We use the same definition for Q (3.7) and redefine Ω: Ω ≡ hW 2 , (3.22) where the enthalpy is defined to be, h = ρ + P . The only significant change to the remaining primitive variable recovery method is setting the relativistic density, D = 0, the magnetic field, B i = 0, and baryon rest mass density, ρ0 = 0. Otherwise the procedure is identical. 3.4 3.4.1 Characteristics MHD Wave Mathematical Description For the following section, we will be using the notation A to represent a 10 × 10 array and V to represent a 1 × 10 array. In Ant´ on et al.[57], the hyperbolic structure of the general relativistic MHD wave equations was investigated. That work is based on the earlier study by Anile [54] for the general relativistic hydrodynamic hyperbolic structure. To determine the hyperbolic nature 60 3.4. CHARACTERISTICS of the equations, Ant´ on et al.defined a variable 1 × 10 array V = (uν , bσ , P, s), where uν , bσ , P are as defined in Chap. 2, and s is the specific entropy. They write the equations of motion as a quasi-linear system: B AµA B ∇µ V = 0, (3.23) where A, B = 1, ..., 10. The 10 × 10 matrices Aµ are given by Cuµ δ α1 β1 bµ δ α5 β 5 Aµ = ρ hδ µ 0 β9 0µ β11 −bµ δ α2 β2 + P α2 µ bβ2 lα3 µ −uµ δ α6 β6 f α7 µ 0µ β10 uµ /c2s 0µ β12 0µ 0µα4 0µα8 , 0µ uµ (3.24) where α and β with subscripts are unique indices that run over the range 1 . . . 4. Ant´ on et al.further define: bα = gαβ bβ ; (3.25) C = ρ0 h + b 2 ; (3.26) P α2 µ = g α2 µ + 2uα2 uµ ; (3.27) lα3 µ = ρ0 hg α3 µ + ρ0 h − b2 /c2s uα3 uµ /ρ0 h; (3.28) f α7 µ = uα7 bµ /c2s − bα7 uµ /ρ0 h. (3.29) We define for each value of the index µ = 0, 1, 2, 3: 0µ β = [0 0 0 0] ; T (3.30) 0µβ = [0 0 0 0] ; (3.31) 0µ = 0. (3.32) and Let φ(xµ ) = 0 define a characteristic hypersurface of Eqn. (3.23) [54], such that its gradient locally has the same velocity as the traveling wave [132]. For example if the waves are travelling in the 61 3.4. CHARACTERISTICS radial direction the gradient of the hypersurface has the form φµ = (−λ, 1, 0, 0), (3.33) where λ denotes the wave speed. To calculate the wave equations, we define a characteristic matrix by contracting the 4-gradient of the hypersurface φµ = ∇µ φ with the 4 matrices Aµ : α1 Caδ β1 Dδ α5 β 5 Aµ φµ = ρ0 hφβ9 0β11 m α2 β2 l α3 −aδ α6 β6 f α7 0β10 a/c2s 0β12 0 0 α4 0α8 , 0 a (3.34) where a = uµ φµ , (3.35) D = bµ φµ , (3.36) lα3 = lα3 ν φν = φα3 + ρ0 h − b2 /c2s auα3 /ρ0 h + Dbα3 /ρ0 h, (3.37) f α7 = f α7 ν φν = abα7 /c2s − Duα7 /ρ0 h, (3.38) mα2 β2 = (φα2 + 2auα2 ) bβ2 − Dδ α2 β2 . (3.39) 0β = [0 0 0 0] , (3.40) We define T 0β = [0 0 0 0] . (3.41) To calculate the characteristic wave velocities we calculate the roots of the characteristic equation defined by the determinant of the characteristic matrix Eqn. (3.34). This leaves us with the equation det Aµ φµ = Ca2 A2 N4 = 0, (3.42) 62 3.4. CHARACTERISTICS where A = N4 = Ca2 − D2 , ρ0 h 1 b2 − 1 a 4 − ρ0 h + 2 2 cs cs (3.43) a2 φ2 + D2 φ2 . (3.44) C is defined in Eqn. (3.26) and a is defined in Eqn. (3.35). We wish to calculate the waves for the MHD in their physical directions along a coordinate xi . To do this, we choose a form for the characteristic hypersurface, φµ = (−λ, 1, 0, 0), (3.45) φµ = (−λ, 0, 1, 0), (3.46) φµ = (−λ, 0, 0, 1), (3.47) for the radial, polar, and azimuthal waves respectively. By substituting these expressions into Eqn. (3.42) we have the characteristic polynomial for waves traveling in the respective directions [57]. By using Eqns. (3.45), (3.46) or (3.47), separately, for φµ in Eqns. (3.35), (3.43) and (3.44) we obtain the characteristic velocities for the MHD waves for each of the spatial directions. We describe the different MHD waves in detail in the following section. 3.4.2 MHD Waves and Characteristic Velocities One of the fundamental properties of our expression of the ideal MHD equations of motion is the exposure of the dependence of the system on the waves of the different constituent fields generated by the flow. For our system these consist of a matter wave, sound waves, two Alfv´en waves, and four magnetosonic waves, which come in pairs, i.e. there are two fast waves and two slow waves so named due to their relative magnitudes when compared to the Alfv´en wave speed. We have explicit calculations for the eigenvalues for the Jacobian matrices for ideal MHD [47]. We list the results here for the waves travelling in the coordinate xi -direction. Matter waves propagate disturbances in the density of the fluid at the speed of the fluid. These waves are stationary in the frame of the fluid and are also called contact discontinuities, meaning the fluid variables are smooth but there is a jump in the density. The speed of the matter waves, 63 3.4. CHARACTERISTICS λM , are found by evaluating the a = 0 root for Eqn. (3.42), i.e. setting (3.35) to zero: λM = αv i − β i . (3.48) When taken to the flat spacetime limit in Cartesian coordinates, (α = 1, β i = 0), this wave is dragged along by the fluid. Sound waves are longitudinal velocity perturbations that propagate parallel to the magnetic field. They result from a small disturbance in the density field of the fluid medium, such as a local compression. The sound waves propagate this disturbance through the fluid medium [133]. Like sound waves, magnetosonic waves are longitudinal, parallel to the velocity of the fluid. There are two types of magnetosonic waves, the fast and slow magnetosonic waves. The difference between the fast and slow waves is the relative motion of the wave to the magnetic field lines. When the magnetosonic wave is perpendicular to the magnetic field lines the magnetic field acts to enhance restoring forces found in sound waves and thus the presence of the magnetic field acts to enhance the speed of the wave [134]. The slow magnetosonic waves are also parallel to the fluid flow; however, these waves are parallel to the magnetic field. The magnetosonic waves are also known as magneto-acoustic waves. The fast and slow magnetosonic waves were originally investigated by Spitzer in 1962 in a Newtonian system [97]. These waves develop due to a nonlinear coupling between the Alfv´en waves and the sound waves. This coupling is seen in the dispersion relation for this particular set of fields, as discussed in section 3.4.1. In the absence of the magnetic field the magnetosonic waves reduce to the sound wave. The magnetosonic wave speeds, λMS , are the roots of the fourth degree polynomial (3.44) in (3.42), N4 = 0: λMS = N4 = ρh 1 b2 4 − 1 a − ρ h + 0 c2s c2s a2 G + B 2 G = 0, (3.49) where a = W −λ + αv i − β i , α (3.50) B = bi − bt λ, (3.51) 1 −(λ + β i ) + α2 γ ii , α2 (3.52) G = 64 3.4. CHARACTERISTICS and cs is the speed of sound as described in Chap. 2, Eqn. (2.57). Like the magnetosonic waves, the Alfv´en waves also arise from the coupling of matter to the magnetic fields. In the ideal MHD limit, as the fluid flows, it drags the magnetic field with it. Since we have compressible flow and the frozen-in condition, the magnetic fields will compress, and will push against each other. This push is the magnetic pressure. The pressure within the magnetic fields eventually overcomes the compression of the fluid and the density of the fluid begins to decrease. This pressure will then exceed the original compression where the fluid will continue to compress the magnetic field and start the process all over again. The resulting oscillatory behaviour is the origin of the Alfv´en waves. These waves result from perturbations that are perpendicular to the fluid velocity and parallel to the magnetic field, and are sometimes known as intermediate waves due to their relation to the magnetosonic waves. The Alfv´en wave speeds, λa , may be calculated by setting A, Eqn. (3.43), to zero: λa− = λa+ = bi + ρ0 h + b 2 u i bt + ρ0 h + b 2 u t bi − ρ0 h + b 2 u i bt − ; (3.53) . (3.54) ρ0 h + b 2 u t Ultimately, when studying an MHD system with hyperbolic divergence cleaning, (Sec. 2.6.1), two more waves arise from treating the numerical divergence violations as fluxes for the auxiliary variable Ψ in Eqn. (2.113). The wave velocities for the auxiliary field are fixed parameters, and for our system are set to ±1, the speed of light15 . The wave speeds calculated in Eqns. (3.48) to (3.54) are also the eigenvalues of the Jacobian matrix in Eqn. (3.3). While we define the Jacobian as the derivative with respect to the conservative variable, the relativistic flux variables are given as a nonlinear combination of conservative and primitive variables, and the conservative variables are a known function of the primitive variables. Calculating the Jacobian is not a one-step process. To find the eigenvalues, we first reformulate the Jacobian matrix [10]: Ai = ∂f i ∂f i = ∂q ∂p ∂q ∂p −1 . (3.55) Recall that we define q to be the conservative variables, f i to be the flux contributions, and p to be the primitive variables. To calculate the eigenvalues and eigenvectors numerically we make the 15 Recall that we set c=1. 65 3.4. CHARACTERISTICS following definitions: A0 ≡ ∂q , ∂p (3.56) Ai1 ≡ ∂f i , ∂p (3.57) Ai1 − A0 λi xi = 0. (3.58) and solve the generalized eigenvalue problem, The superscript i denotes the direction of the eigenvalue equation. For example, to calculate the radial eigenvalues we use the radial fluxes in Eqn. (3.57) and solve the eigenvalue problem in Eqn. (3.58) with i = r. We solve Eqn. (3.58) three times, once for each dimension. In the case of symmetries we will have a trivial solution, λ = 0 for all waves, since net fluxes in a symmetric direction will be zero, we do not need to solve (3.58) for the symmetric dimension(s). The eigenvalues, λi , of (3.58) are no different than those for the system Ai y = λi y, (3.59) where Ai is defined in Eqn. (3.55). We relate (3.58) and (3.59) by [10], Ai1 − A0 λi x = 0, (3.60) i Ai1 A−1 0 − Iλ A0 x = 0, (3.61) where I is the identity matrix. Then the eigenvectors, x, of (3.58) are related to the eigenvectors, y, of (3.59) by, A0 x = y. (3.62) The eigenvectors y are the eigenvectors for the system of interest. There is no known closed form solution for the eigenvectors for the GRMHD system; thus, we use LAPACK [135] to perform the spectral decomposition16 . We solve the eigensystem (3.58) and (3.62) for each spatial dimension, i. In the event of a symmetry the number of eigensystems solved is reduced. For example in the 16 Namely we used the generalized eigenvalue and eigenvector solver DGGEV. 66 3.5. CONSERVATIVE METHODS Qni Qni+1 xi−1/2 xi+1/2 xi+3/2 Figure 3.1: A graphical representation of a finite volume. Qi denotes the state vector, averaged over a cell defined by xi−1/2 ≤ x ≤ xi+1/2 . The flux is assumed to enter or leave a cell, Qi , through the cell interface at xi+1/2 . case of axisymmetry we do not need to calculate the i = φ equations. 3.5 Conservative Methods When there are discontinuities in the initial data set, solving equation (3.1) requires special treatment for the discretization of the equations of motion. An older approach used finite differencing of the equations with additional terms to introduce artificial viscosity17 ; however, we pursue a less diffusive technique that is specially developed for conservation laws and discontinuities in the domain, the high resolution shock capturing (HRSC) algorithms. HRSC methods have the advantage of ensuring the conservation of the variables q, and that discontinuities such as shocks are well resolved. To capture the discontinuities that may develop in a fluid flow, we use an integral form of the conservation equations, which is known as the weak solution. We consider Eqn. (3.1): ∂t q + ∂i f i (q) = s(q). (3.63) Here q are the conserved variables, f i (q) is the flux density vector, and s(q) are the source terms. 17 This form of solution is not treated here, for more details we refer the reader to [8, 28, 29] 67 3.6. THE RIEMANN PROBLEM As shown in Fig. 3.1, we integrate this equation over a finite volume ∆Vi with the domain xi−1/2 ≤ x ≤ xi+1/2 , ∂t q + ∂i f i (q) dV = (3.64) sdV. ∆Vi ∆Vi Using Gauss’ Theorem, f i dSi , ∂i f i (q) = (3.65) Si ∆Vi we can re-write the integral over the flux terms, ∂ ∂t f j (q)dSj = q+ ∆Vi (3.66) sdx. ∆Vi Si This equation shows us that the change in time for the quantity q in a volume Vi is from the flux through the volume’s surface, as well as from the source within Vi . Now, to finalize the integral form we integrate over a time interval tn ≤ t ≤ tn+1 to get tn+1 tn ∂ ∂t i tn+1 q(t, x )dV + ∆Vi tn+1 i s(q(t, xi ))dSj dt. f (q(t, x ))dSj dt = tn tn (3.67) ∆Vi Equation (3.67) is the preferred mathematical form of a conservation laws, since it does not depend on continuity of the state vectors. This means that in the presence of a discontinuity, the integral form will still be satisfied [36]. The finite volume method described above involves the use of approximate Riemann solvers to handle discontinuities in the fluid variables. To understand the approximate methods it is beneficial to understand the Riemann problem and the exact solutions. In the following two sections we first discuss the Riemann problem in general along with the possible solutions, then the exact Riemann solvers implemented by Mart´ı [49] and Giacomazzo [52]. 3.6 The Riemann Problem The most elementary problem involving discontinuities, as illustrated in Fig. 3.2, has two distinct constant states separated by a discontinuity. This is known as the Riemann problem. For simplicity, we restrict our attention to a problem in one spatial dimension. In Fig. 3.2 we have an example of a a discontinuous initial data set for the state vector q = (q1 , q2 , ..., qp ), which is used for the conservation equation (3.1). The initial data are expressed as, 68 3.6. THE RIEMANN PROBLEM q qL qR x=0 x Figure 3.2: A scalar example of an initial data set for the Riemann Problem. The qL and qR are constant state vectors. Each of these vectors are not a function of the coordinates and are generically separated by a discontinuity at the origin at t = 0. When the divider is removed, Riemann’s solution describes the resulting interaction of the two states. q(0, x) = qL , x < 0; (3.68) qR , x > 0, where qL and qR are distinct constant values. If we were to study the advection equation as seen in Eqn. (1.1), with s = 0 and v = Const., we have the simple solution qL , x − vt < 0, q(t, x) = qR , x − vt > 0. (3.69) Any information on the left of the discontinuity seen in Fig. 3.2 would propagate upstream with a fixed velocity v, likewise for the information on the right of the discontinuity. In the advection equation example the characteristic line x − vt = 0 denotes the location of the discontinuity for all time. The Riemann problem becomes more complicated in equations where the fluxes are nonlinear, such as in hydrodynamic systems of hyperbolic equations. In particular, considering our mag69 3.6. THE RIEMANN PROBLEM t λFast MS λa q1 qL λSlow MS q2 λM q3 q4 λSlow MS q5 λFast MS λa q6 qR x Figure 3.3: The general magnetohydrodynamic characteristic fan. We see the two Alfv´en waves, Slow λa , the fast λFast MS and slow λMS magnetosonic waves, and the contact discontinuity λM . These waves decompose the Riemann problem into 8 regions denoted qL , q1 → q6 , and qR . The treatment of the different regions for an exact solution is described in [52]. When solving the Riemann problem either exactly or approximately the region around the contact discontinuity are typically referred ∗ to as star-regions and are denoted qL∗ and qR on the left and right, respectively. This figure is a replica of that seen in [52]. netohydrodynamic system, we expect to have a minimum of eight characteristics. If our system is strictly hyperbolic each characteristic will be distinct. The set of characteristics for a (magneto)hydrodynamic system are known as a characteristic fan. We refer the reader to figure 3.3 for a sample characteristic fan. With our system we have two extreme waves denoted λMSFast . Any part of the domain to the left of the left-most λMSFast will simply be the initial data qL , and the domain to the right of the right-most λMSFast will be qR . The region between these waves is often referred to as the star region and is denoted q∗ . The solution in this region is complicated by the interaction of the different waves described in Sec. 3.4. The simplest interactions of the different waves come in three general forms. The first we will define and discuss are shock waves. Then we will discuss rarefaction waves, and finally contact discontinuities, which always arise from the presence of matter waves λM . For the next several sections, for simplicity, we assume that the conservation vector q is comprised of one element, q = q(t, x). As we have seen, the advection equation has a constant flux 70 3.6. THE RIEMANN PROBLEM and is therefore not suitable to use as an example for more complicated waves. We use an illustrative example of the shock and rarefaction solutions by using the one dimensional inviscid Burger’s equation [19], ∂t q + ∂x q2 = 0. 2 (3.70) The Burger’s equation has characteristics λ = q. 3.6.1 Shocks One class of solution to the Riemann problem are known as shock waves. Shocks are defined as locations where characteristic lines converge. To produce a shock in Burger’s equation (3.70), we use initial data (3.68) with qL > qR . We see that in this case the characteristics on the left move faster than those on the right, and thus the characteristic lines will converge. An example of a shock forming from discontinuous initial data can be seen in Fig. 3.4, where the shock (green line) originates at x = 0 with slope s = 0.5. Physically, shock waves are thin transitions from supersonic to subsonic flow which involve compression and dissipation [136]. A propagating wave in an ideal compressible fluid may lead to an infinite gradient within a finite amount of time. The part of the domain where the gradient blows up is referred to as a shock. The fluid close to either side of the shock may be considered uniform due to the thinness of the shock region. Shock waves have velocities that may be determined by the so-called Rankine–Hugoniot jump condition, which are the conditions that relate the flux of the state quantities across a discontinuity [97, 19]. For relativistic fluid systems the Rankine–Hugonoit jump conditions are described in detail in Majorana [137], or Anile [54]. The Rankine–Hugoniot jump condition for an ideal fluid considers a steady flow across a jump in initial data, demanding a conservation of baryon density Eqn. (2.47), and energy-momentum Eqn. (2.48). We consider the integral solution (3.67); if we let the the interval of integration be small enough such that the variation of the variables and flux over the integrals is small we can write ∆x qL − ∆x qR = ∆t f (qR ) − ∆t f (qL ) + O(∆t2 ). (3.71) The term O(∆t2 ) includes the variations of the conserved quantities and the flux. Looking at an infinitesimal time step, ∆t, we have a velocity, s = dx dt . The velocity may be re-written as 71 3.6. THE RIEMANN PROBLEM Figure 3.4: An example of a shock wave characteristic diagram for Burger’s equation ∂t q + ∂t (q 2 /2) = 0, with initial data q(x, 0) = 0.8 if x < 0; q(x, 0) = 0.2, if x ≥ 0. In this case the shock wave moves with speed, s, determined by the Rankine–Hugonoit jump conditions. For the Burger’s (qL ) = 0.5, which is evident since the characteristic lines converge on equation we have, s = f (qqRR)−f −qL the t = x/s line (green). 72 3.6. THE RIEMANN PROBLEM ∆x = s∆t. This leaves us with s(qL − qR ) = f (qL ) − f (qR ), (3.72) from which the shock speed, s, may be determined. This is the only allowable shock speed. In general the shock speed will be a function of time; however, in this development the time step was assumed to be small enough that we can ignore this variation. Using this information we then have the solution for the Burger’s equation with discontinuous initial conditions that lead to a shock wave: qL , x − st < 0; q(t, x) = qR , x − st > 0. (3.73) For the Burger’s equation the shock speed (3.72), is determined to be, s= 3.6.2 2 1 qL2 − qR 1 = (qL + qR ) . 2 qL − qR 2 (3.74) Rarefaction Waves Another class of solution to the Riemann problem are known as rarefaction waves. The characteristics of a rarefaction wave emanate from the discontinuity, spreading out, as illustrated by the blue lines that originate at the origin in Fig. 3.5. Rarefaction waves expand with time, reducing pressure, and usually follow in the wake of a shock wave. For a rarefied wave solution to Burgers equation we use initial data as seen in Eqn. (3.68), with qL < qR . We see that in this case the characteristics on the left move slower than those on the right, and thus the characteristic lines will diverge. If we attempt to use the Rankine–Hugoniot jump conditions with these initial data the result will be shock waves emanating from the discontinuity. This solution is both unphysical and mathematically unstable. For a discussion on the unstable solution we refer the reader to [19, 20]. To derive the physical solution, we change the initial data to introduce a smooth interpolated region connecting the left and right data. We depict the modified data in Fig. 3.6. The initial data are now: 73 3.6. THE RIEMANN PROBLEM Figure 3.5: A rarefaction wave characteristic diagram for Burger’s equation ∂t q + ∂t (q 2 /2) = 0, with initial data q(x, 0) = 0.2: if x < 0; q(x, 0) = 0.8, if x ≥ 0. In this case the characteristic structure obeys q(x, t) = 0.2, if x < 0, q(x, t) = x/t if 0 ≤ x ≤ t, or q(x, t) = 0.8, if x ≥ 0.8t. The rarefaction is coloured blue. qR qL xL x=0 xR Figure 3.6: We change the initial data for a rarefied wave. Instead of a sharp discontinuity at the origin we introduce interpolated data connecting the two sides. We recover the original initial data by taking the limit as xL → 0 and xR → 0. 74 3.6. THE RIEMANN PROBLEM qL , q(0, x) = qL + qR , x ≤ xL ; qL −qR xL −xR (x − xL ), xL < x < xR ; (3.75) x ≥ xx . The wave on the left side is the head of the rarefaction wave, and is governed by the usual characteristic equation, x = xL + λ(qL )t, (3.76) while the right is the tail of the rarefaction wave with the characteristic equation, x = xR + λ(qR )t. (3.77) The full solution of the system is then: q(t, x) = qL , x − xL < λL t; x−xL t , λL t < x − xL < λR t; qR , x − xR > λR t. (3.78) This is the solution to the interpolated problem. To get the solution to the original problem with the discontinuity at the origin, we take the limit xL → 0 and xR → 0. The structure of the rarefaction solution is unaltered in this limit [20]: qL , x < λL t; x q(t, x) = λL t < x < λR t; t, qR , x > λR t. (3.79) This solution is both physical and stable [20]. We may interpret this solution as the value of q being all solutions between qL and qR at the discontinuity. Thus the characteristics λ take on all values between λL and λR . Larger values propagate faster than lower values, so the discontinuity degrades immediately to a centred rarefied wave [20]. 75 3.6. THE RIEMANN PROBLEM Centred rarefied waves are also known as similarity solutions. Similarity solutions have the form: q(x, t) = qˆ x = qˆ (˜ x) , t (3.80) and are constant along any ray x/t = const. Using the similarity solution we substitute Eqn. (3.80) into the linearized conservation equation (3.2) to get ∂t q(t, x) + df df 1 ′ x ∂x q(t, x) = − 2 qˆ′ + qˆ = 0, dq t dq t (3.81) where qˆ′ = ∂ qˆ/∂ x ˜. Equation (3.81) has two possible solutions, qˆ = Const, (3.82) df x (ˆ q (˜ x)) = . dq t (3.83) or Using Eqn. (3.83) we can determine the solution for a centred rarefaction wave [19]. 3.6.3 Contact Discontinuities The last family of solutions to the Riemann problem that we discuss are known as contact discontinuities. A contact discontinuity is defined by an abrupt jump in fluid density with no pressure gradient across the jump. When a contact discontinuity forms it is stationary in the fluid’s reference frame. Contact discontinuities arise when the characteristics for the system are parallel, neither diverging nor converging, and do not create entropy, but rather separate regions of different entropic value. Due to the fact that contact discontinuities separate the regions of different entropy they are often referred to as entropy waves, since they carry information about the entropy increase. Physically, these will occur when two opposing shocks intersect [138]. These are all the possible basic waveforms for the simple Burger’s equation. When studying systems with nonlinear fluxes, combinations of these waves may exist in a single problem. This is typical in shock tube tests studied in hydrodynamics. We discuss these further in Chap. 4 when we apply tests to our code. An example of such a solution may be seen in Fig. 4.3. We now briefly describe the numerical solution of the exact Riemann problem. 76 3.7. THE RIEMANN PROBLEM: EXACT SOLUTIONS 3.7 The Riemann Problem: Exact Solutions We present a simplified description of the exact solution here, a full description may be found in [52]. The exact solution to the special relativistic hydrodynamic Riemann problem was first found by Mart´ı et al.[49], followed by Giacomazzo et al.[52], who derived the exact solution for special relativistic magnetohydrodynamics. To solve for the magnetohydrodynamic system of equations exactly, they used the shock and rarefaction behaviour of the system of equations at each the discontinuities. If the discontinuity is a shock the solution is determined by a relatively simple set of algebraic expressions. In the event of a rarefaction wave the solution is more complicated since they were required to solve a system of ODEs. The final step is to use matching conditions to connect the fluid variables across the discontinuity. In Fig. 3.3 only λM will be a contact discontinuity. The matching conditions will determine the value of the intermediate state, q∗L = q3 , q∗R = q4 , that exists on either side of the contact discontinuity seen in Fig. 3.3. Giacomazzo et al.call this approach the Pt -method, since the intermediate state and the determination of the discontinuity type (shock or rarefaction) is dependent only on the total pressure (magnetic pressure plus thermal pressure). They switch methods when they come to the contact discontinuity due to the complicated nature of MHD waves. By continuing to use the total pressure they find that the methods needed to solve the matching conditions are too sensitive to the initial guesses. To rectify this they specify the values of the variables on either side of the contact discontinuity in terms of the transverse components of the magnetic field. For complete details including the system of equations used for the exact special relativistic MHD method we refer the reader to [52]. The exact solver is computationally expensive [51, 139], so to perform a parameter survey more efficiently we use approximate Riemann solvers, as discussed in Sec. 3.8.3. The exact solver is useful for testing how well approximate Riemann solvers capture shocks. We do not implement our own exact solver, but use Giacomazzo et al ’s solver to test the shock-capturing capabilities of our code. The results from these tests may be seen in Chap. 4. 3.8 The Godunov Method Methods that break down the domain into finite volumes and treat the interface between each volume as a Riemann problem are known as Godunov methods. We give a general overview of 77 3.8. THE GODUNOV METHOD tn+1 tn xi−1 xi xi+1 Figure 3.7: A sample one dimensional discretization. The temporal discretization is denoted by the current time step, tn , and the advanced time step tn+1 . The spatial discretization denotes the current spatial location, xi . The solid vertical lines denote a cell interface. the Godunov method below, and the specific details are described in Sec. 3.8.1. For simplicity we outline the problem using two dimensions, such that, q = q(t, x). We use the discretization notation described in Chap. 1, and seen in Fig. 3.7, such that the current discrete time level is denoted, tn and the advanced time level is denoted tn+1 . The spatial discretization is denoted xi . The cell centers are located at xi , while the cell interfaces surrounding that cell are located at xi±1/2 . To solve Eqn. (3.67) using Godunov-type methods, we start by dividing the computational domain, V (of arbitrary spatial dimension), into finite volumes, ∆Vm , such that V = m ∆Vm . The next step is to define cell-averaged quantities, Qni at the cell centres for time level tn : Qni ≡ 1 ∆V q(tn , x)dV . (3.84) ∆V Then we extrapolate the variables at the cell centre to the cell interface. In Fig. 3.1 this would require that we extrapolate the data from cell centre Q(tn , xi ) to the right side of the interface located at xi+1/2 , denoted QR (tn , xi+1/2 ), and the values at Q(tn , xi+1 ), to the left side of the cell interface located at xi−1/2 , denoted QL (tn , xi−1/2 ). The techniques used to perform the interpolation are discussed in Sec. 3.8.2. We now focus on discretizing the surface integral using the surface of the cell. The temporal integrals over the flux integrals are not, in general, easily solved. Godunov [23] proposed a first order method to approximate the flux integrals by replacing the conservative quantities Q(t, x) with piecewise constant functions. His approach treats each cell interface as an individual, or local, Riemann initial value problem. As long as the waves from adjacent cells do not interact, each local Riemann problem can be solved exactly. The solutions to the local Riemann 78 3.8. THE GODUNOV METHOD n problems are treated as constant in time, which allows us to approximate the flux integrals, Fi+1/2 : tn+1 f j (q(t, xi+1/2 )dSj dt = ∆t tn Fji+1/2 dSj . (3.85) We discretize the surface integrals in Eqn. (3.67) and (3.85), F j dSj = F j ∆Sj , (3.86) ∆Sj for an arbitrary cell index. The sum is over all surfaces bounding the cell. We approximate the source integral as s(q(tn , xi ))dxdt. ˆsni ∆x∆t = ∆t (3.87) ∆V We perform the temporal integral in (3.67), ∆t dQ(t, xi ) = Q(tn+1 , xi ) − Q(tn , xi ) = Qn+1 − Qni , i dt (3.88) where tn+1 denotes the advanced time step. We then write the complete discretized form of (3.67) in one dimension using Eqns. (3.84), (3.85), (3.86), (3.87), and (3.88) to get, Qn+1 = Qni − i ∆t Fi+1/2 − Fi−1/2 + ∆t sˆ, ∆x (3.89) where in one dimension in Cartesian coordinates ∆S = 1, and ∆V = ∆x. Eqn. (3.89) is the final discretized form for the Godunov method. For full details on the Godunov method we refer the reader to LeVeque [19] or Font [21]. The logical flow of the Godunov method and the generalized methods based on the Godunov method are; • First, using the cell-averaged quantities, we construct a piecewise constant function, used to approximate the value of the cell at each interface depending on the values of surrounding cells. In the literature this is known as “reconstruction”. • Next, we solve the Riemann problem at the cell interfaces. • Finally, we time evolve the state Q(tn , x) to the solution Q(tn+1 , x) using an explicit numerical 79 3.8. THE GODUNOV METHOD integration technique. This is the basis of the Godunov method, and as long as we adhere to these steps, we may modify any of them to extend Godunov’s method to higher order schemes. The benefits of Godunov’s method are that this method is stable, conservative, and handles shocks. There are two drawbacks to using Godunov’s original method; first, by itself, this method is only first order accurate, and the second drawback is that the solution of the full Riemann problem may be computationally expensive, particularly when studying general relativistic magnetohydrodynamics. To circumvent these limitations we consider more sophisticated high-resolution shock-capturing methods. To increase computational efficiency of the method, we introduce approximate Riemann solvers. We outline three such methods in section 3.8.3: the Roe solver; a relativistic Harten–Lax–van Leer (HLL) solver; and the Lax–Friedrichs method. We now explicitly describe the relativistic form used for the Godunov-type schemes, they are more complicated than the method outlined above. 3.8.1 The Relativistic Godunov Scheme Due to the subtleties when extending the Godunov scheme to relativistic systems, we outline the method here. In (2.107) we saw the set of nonlinear conservation laws for general relativistic magnetohydrodynamics are written in general as 1 1 ∂√ ∂ √ √ γ q+ √ −g f i (q) = s(q), −g ∂t −g ∂xi (3.90) where g is the determinant of the 4-metric, and γ is the determinant of the 3-metric as defined in Chap. 2. q is the vector of conservative variables, f is the set of fluxes for the corresponding conserved variable, and s is the vector of source terms also defined in Chap. 2. We require the integral form of this equation, so just as in the general method presented in Sec. 3.8.1 we integrate Eqn. (3.90) over some small cell or volume, ∆V , of spacetime, ∆V 1 ∂ √ √ ( γ q) dV (4) + −g ∂t ∆V 1 ∂ √ √ −g f i (q) dV (4) = −g ∂xi s(q)dV (4) , (3.91) ∆V 80 3.8. THE GODUNOV METHOD (i, j + 1) dΣj+1/2 yj+1/2 ∆Vi,j (i − 1, j) dΣi−1/2 (i, j) dΣi+1/2 (i + 1, j) yj−1/2 dΣj−1/2 (i, j − 1) xi−1/2 xi+1/2 Figure 3.8: The 2D cell for the finite volume method. The cell volume ∆Vi,j = (x1i+1/2 − x1i−1/2 )(x2j+1/2 −x2j−1/2 ), is defined for the cell containing the point (i, j) with a cell surface elements dΣ bounding the cell as labeled above. The flux of the conservative equations are calculated at the centre of the cell interfaces, denoted by the location of the arrows. 81 3.8. THE GODUNOV METHOD where dV (4) = √ −gdx0 dx1 dx2 dx3 . For relativistic systems the cell average is defined to be [21], Q≡ with ∆V (3) = 1 ∆V (3) q∆V (3) , (3.92) ∆V √ γ∆x1 ∆x2 ∆x3 . We re-write the first term in Eqn. (3.91) as, ∆V 1 ∂ √ √ ( γ q) dV (4) = ∆V (3) −g ∂t ∆x0 ∂ Q∆V (3) dx0 . ∂t (3.93) Next, using the divergence theorem, we re-write the second term in Eqn. (3.91) as, ∆V 1 ∂ √ √ −g f i (q) dV (4) = −g ∂xi f (q)dΣdx0 , ∆x0 (3.94) ∂V where dΣ is normal to the surface of the finite volume ∆V . We approximate the surface integral, f j (q)dΣj = Fj ∆Σj (3.95) ∂V where the sum is over all surfaces bounding the volume element. We define ∆Σj = √ −gdxi dxk ǫijk such that the direction of the surface element is outward to the surface area in question. ǫijk = √ −g[ijk] is the relativistic Levi–Civita totally antisymmetric tensor. We have also defined Fi (q) = 1 ∆x0 f i (q)dx0 , (3.96) ∆x0 the time averaged flux. Our last step with respect to the spatial integrals, is to approximate the integral of the source term, (4) ˆ S(q)∆V dx0 , S(q)dV (4) = ∆V (4) ˆ where we write S(q)∆V = ∆V (3) (3.97) ∆x0 √ S(q) −gdx1 dx2 dx3 . Now we approximate the time integration (3) n (Q∆V (3) )n+1 )i,j + ∆x0 i,j − (Q∆V ˆ n ∆V (4) , ∆Σi Fij (q) = S i,j (3.98) j and assume that the evolution is over such a short period of time that the sources and fluxes remain constant in time. The sum over j in (3.98) is the sum of the fluxes over all cell faces, dΣi+1/2 , dΣi−1/2 , dΣj+1/2 , dΣj−1/2 as seen in Fig. (3.8). When considering stationary spacetime 82 3.8. THE GODUNOV METHOD metrics, the volume elements are independent of time, so we can easily write the final form of the approximate solution of Eqn. (3.90), n Qn+1 i,j = Qi,j − ∆x0 ∆V (3) (4) ∆V ˆ ∆Σi Fik (q) + S(q) . ∆V (3) i,k (3.99) The quantity Qn+1 is the advanced time value of the cell average quantity for cell (i, j). To i,j handle the flux time integral in Eqn. (3.96) without explicitly performing the integral, we rely on approximations such as the Roe solver, Harten–Lax–van Leer, or the Lax–Friedrichs methods. These methods are described in Sec. 3.8.3. In the remaining parts of this section, we describe the different cell interface reconstruction techniques used, followed by the flux approximations used to solve the system described by Eqn. (3.99), also with implementation notes. We then describe the integration methods used and the different boundary conditions used. We end this chapter with a discussion of other numerical techniques that are often needed to ensure a viable solution. 3.8.2 Variable Reconstruction at Cell Boundaries Before we can evaluate the flux approximations described above, we require the values of the state vectors, q, at the cell interface. First we introduce the piece-wise constant method used by Godunov. To obtain the values of the state vectors for our system we use the piecewise linear method18 for the cell reconstruction. This method typically uses one of two techniques to prevent spurious oscillations from appearing during the reconstruction; the “minmod limiter” or the “monotised central limiter”.19 Piecewise Constant Method The piecewise constant method is the simplest way to reconstruct the cell interface. Consider Fig. 3.1: to reconstruct the cell value on the left and right of the interface at xi+1 , we use, qLi+1/2 = qi (3.100) qR i+1/2 = qi+1 . (3.101) 18 We are aware of the higher order piecewise parabolic method [140], but chose not to implement it in this study. techniques do exist but are not as commonly used in relativistic studies. We refer the reader to LeVeque [19] for a detailed discussion of the existing methods. 19 Other 83 3.8. THE GODUNOV METHOD This method was used by Godunov in his original prescription [23] and results in first order accurate solutions. To increase spatial accuracy, higher order techniques are used for the cell reconstruction, but all of them revert to the piecewise constant method when reconstructing cells that are in the presence of extrema and shocks. Piecewise Linear Method The algorithm for piecewise linear method looks for the minimum slope within surrounding cells to extrapolate the value of the state vector q at the cell centre to the cell interface. By choosing the minimum slope to extrapolate the variables we smooth the data used to calculate the fluxes while minimizing the possibility of introducing unphysical oscillations into the system. Minmod: Looking at Fig. 3.9, when using this method to reconstruct a cell interface, we require information of the states in four surrounding cells. To find the reconstructed state at the interface xi−1/2 , the left interface of cell qi , we use the cells labeled qi−1 , qi , qi+1 to calculate the slopes s2 , s3 , then uses the minmod algorithm (Eqn. (3.102)) to determine which slope is a minimum of the two. The line with the minimal slope is used to reconstruct the variable at that interface. The concise algorithm is shown in Eqn. (3.102). In the event that we are reconstructing a state near an extremum, the minmod algorithm returns a zero slope which indicates that the reconstruction reduces to the piecewise constant method described above. Likewise, to find the reconstructed value of the xi−1/2 interface for the cell qi−1 , we use cells qi−2 , qi−1 , qi to find slopes s1 , s2 , then use the minmod limiter to select the slope that results in a value that is not a new extremum: a, if |a| < |b|; minmod(a, b) = b, if |b| < |a|; 0, if ab ≤ 0. (3.102) The limiters are introduced to prevent the oscillations that will occur when the reconstruction produces new local maxima. Referring to Fig. 3.9, we see that the reconstructed value qL at the coordinate point xi−1/2 is calculated by qL = qi−1 + σ1 (xi−1/2 − xi−1 ), (3.103) 84 3.8. THE GODUNOV METHOD s3 s2 3 1 qi 2 qi+1 qi−1 qi−2 s1 xi−2 xi−1 xi−1/2 xi xi+1 Figure 3.9: To reconstruct the value at the cell face denoted xi−1/2 from the left, we use the cells qi−2 , qi−1 to calculate the slope of the line s1 , and qi−1 , qi to calculate the slope of the line s2 . When we have the slope we determine which of the two has the minimum slope. The line with the minimum slope is used to calculate the quantity qi−1/2 of the left side of the cell interface xi−1/2 . In the diagram above, the quantity qi−1/2 is determined at the point labeled 2, as calculated using the line labeled s2 . To calculate the cell face xi−1/2 from the right, we follow a similar procedure using the cell values qi−1 , qi , qi+1 . In this case, the slopes of the lines s2 and s3 are of opposite signs, so we take the value of the cell qi to be the value of the cell interface from the right, to prevent the production of local extrema in this method. This special case occurs near shocks and extrema, which reduces the order of the method to first order. See the text for a detailed description. 85 3.8. THE GODUNOV METHOD and the value qR at the same coordinate point is qR = qi − σ2 (xi−1/2 − xi ), (3.104) where σ1 = minmod(s1 , s2 ) and σ2 = minmod(s2 , s3 ). In this particular configuration, we see that if we were to consider linear extrapolations alone, we are at risk of producing a non-monotonic value such as the point labeled 3 in Fig. 3.9. The minmod limiter will reconstruct that value by using the piecewise constant reconstruction value qi for that cell. Monotized Central Difference: Instead of using the minmod limiter we can use the monotized central difference (MC) limiter. This method acts with the same arguments, but is modified such that it does not reduce the slope as dramatically as the standard minmod method: MC(a, b) = Here 2a, if |a| < |b| and 2|a| < |c|; 2b, if |b| < |a| and 2|b| < |c|; c, 0, (3.105) if |c| < 2|a| and |c| < 2|b|; if ab ≤ 0. c= (a + b) . 2 We compare the minmod and MC algorithms in chapter 4, Figs. 4.6 and 4.7. With the variables reconstructed at the cell interfaces, we can now calculate the flux across the cell interface, where we perform the flux approximations. 3.8.3 Flux Approximations There are several different techniques used to approximate flux calculations in Eqn. (3.96), to replace the exact Riemann solver. The techniques we used through this research are the Roe solver, Harten–Lax–van Leer, and Lax–Friedrichs. We describe these methods below. For simplicity we assume the system we are approximating is one dimensional, and expressed in Cartesian coordinates. All flux approximations will be calculated at the cell interfaces, e.g. xi+1/2 , and so we drop the grid index for notational simplicity. 86 3.8. THE GODUNOV METHOD Roe’s Method Roe’s linearized solver approximates the conservation equations by using the linearized form, ∂t q + ∂i f i (q) = 0 −→ ∂t q + Ai ∂i q = 0, (3.106) where Ai = ∂f i /∂q is the Jacobian for the ith direction flux vector. Roe’s method treats the Jacobian for the system as a constant Jacobian matrix. This matrix is a function of the data states qL and qR . We will denote the Roe approximate Jacobian matrix as a function of the initial state vectors qL and qR by A¯i = A¯i (qL , qR ). The original system is replaced by an approximate Riemann problem, ∂t q + A¯i ∂i q = 0. (3.107) Since the Jacobian matrix A¯i is treated as a constant the linearized equation can be reduced to a set of scalar advection equations by diagonalizing the Jacobian A¯i = Ri Λi R−1i = Ri Λi Li , R−1i = Li , (3.108) (3.109) where Ri denotes the matrix of right eigenvectors Ri = r1 |...|rp , (3.110) such that Ai rp = λrp for a system of p equations. Likewise, Li denote the matrix of left eigenvectors T (3.111) Λi = diag(λip ) (3.112) Li = [l1 |...|lp ] , such that lp Ai = λlp . We have also defined as the diagonal matrix consisting of the eigenvalues of the Jacobian matrix. The relationship between the right, rp , and left, lp eigenvectors is rj · li = δij . 87 3.8. THE GODUNOV METHOD Now we have ∂t q + RΛR−1 ∂i q R−1 ∂t q + ΛR−1 ∂i q ∂t ω + Λ∂i ω = S(q), (3.113) = R−1 S(q), (3.114) = S(ω), (3.115) where the last expression is a decoupled set of linear equations which each have solution ω p (x, t) = ω p (x − λp t). (3.116) Each variable ω p is known as a characteristic variable and is introduced only to produce a decoupled system of equations. In general, the ω p will not have a direct physical interpretation. The full solution of (3.2) is a linear combination of the solutions in Eqn. (3.116) [19], n qj (x, t) = Rpj ω p , (3.117) p=1 where Rpj denotes the jth element of the pth right eigenvector. Equation. (3.117) is an exact solution to the approximate Riemann problem. For this analysis to hold Roe [141] recorded criteria for the construction of a Jacobian matrix: 1. A¯ must be a linear mapping from q to f i ; ¯ L , qR ) → ∂f /∂q smoothly as qR , qL → q; 2. A(q ¯ L , qR ) · (qR − qL ) = f (qR ) − f (qL ); 3. A(q ¯ L , qR ) are linearly independent (diagonalizable with real eigenvectors). 4. eigenvectors of A(q Here qL and qR are the left and right state vectors across a cell interface. Item two states that the approximate Jacobian must be consistent with the true Jacobian in the appropriate limit. The first, second, and fourth criteria are satisfied by defining the Jacobian matrix at the cell interface as a function of the average of the two states at the cell interface [36], ¯ q ¯ L , qR ) A(q = (qR + qL )/2 ¯ q). → A(¯ (3.118) (3.119) 88 3.8. THE GODUNOV METHOD This setup does not necessarily satisfy the third condition [141], but due to the relative simplicity, it is implemented in relativistic fluid codes [36] also used in the studies by Font et al. [1, 2, 3]. Once the criterion above is met the Jacobian may be treated as a matrix with constant coefficients, so we may diagonalize and solve the system of equations as outlined above in Eqns. (3.108) to (3.117). With the solution of the approximate Riemann problem we can go on to determine the corresponding flux functions, F= 1 1 (f (qR ) + f (qL )) − 2 2 p |λp |∆ωp rp , (3.120) where λp are the eigenvalues of the Jacobian and rp are the associated eigenvectors [19]. ∆ωp are defined by jumps in the fluid variables along the interface, (qR − qL )j = ∆ωi Rij , (3.121) which is easily solved when we invert the matrix of right eigenvectors, Rpj to obtain the matrix of left eigenvectors Ljp = Rpj −1 , −1 ∆ωi = (qR − qL )j Rij . (3.122) We can consider the flux approximation, (3.120), as an arithmetic average for the left and right flux, while adding a stabilizing correction term − 1 2 p |λp |∆ωp rp , (3.123) providing viscous dissipation [19]. The Roe scheme is the most costly of the approximate Riemann solvers we used, due to the fact that it requires a full spectral decomposition. This is particularly true for the general relativistic magnetohydrodynamic systems, where the eigenvectors are not known in closed form as they are in special relativistic hydrodynamics. We use numerical routines such as LAPACK [135] to perform the spectral decomposition. Despite the computational cost, this approximation is the approximate method that is least diffusive; consequently, it captures the discontinuities the best. This point is made clear in Chap. 4 when we use different methods to capture the Kelvin–Helmholtz instability using a setup described by Frank et al.[142] and Stone.20 The other flux approximations used in this work make use of only a few of the wave speeds, 20 http://www.astro.princeton.edu/∼jstone/tests/kh/kh.html 89 3.8. THE GODUNOV METHOD t λ− λ+ q∗ qL qR x Figure 3.10: The characteristic fan for the HLL flux approximation. We only keep the extreme waves, which for the GRMHD case correspond to the fast magnetosonic waves. These waves split the solution into the left qL and right qL states, and a “star” state defined by Eqn. (3.125). ignoring the information contained in the eigenvectors. By neglecting eigenvector information these flux calculations are much faster to compute, but only by sacrificing detailed knowledge of all the waves in the GRMHD system. We also note that in our implementation the Roe method breaks down when the eigenvectors become nearly degenerate, and cannot be used to handle all physical situations; in particular, we found that the method breaks down for extremely low pressure values. Thus in chapter 4 we were unable to use the Roe solver to solve the initial conditions used for Figs. 4.6 and 4.7. HLL The Harten–Lax–van Leer method is much faster than the Roe solver and still allows for a sharp capture of a shock or rarefaction [143]. This is seen in the Riemann shock tube tests performed by [19, 123] as well as our own code verification tests shown in Chap. 4 (i.e. Fig. 4.2). In this approximation, we use the two fastest characteristic velocities of the system to separate the system into three regions. The information about the other waves is not used for the flux approximations. Referring to Fig. 3.3, we treat the information in the regions q1 → q6 as a single state denoted q∗ . The characteristic fan used for the HLL flux approximation is found in Fig. 3.10. The disadvantage 90 3.8. THE GODUNOV METHOD of this method is that it treats all waves travelling slower than the extreme wave speeds with lower resolution. In particular, for fluid systems, all the details for contact discontinuities (entropy waves) are ignored [19]. This solver separates the solution into three states; the left and right (qL , qR ) as well as an intermediate or “star” state q∗ , where qL , if x < λ− t; q(x, t; qL , qR ) = q∗ , if λ− t ≤ x ≤ λ+ t; qR , if x > λ+ t, q∗ = λ+ qR − λ− qL − f (qR ) + f (qL ) λ+ − λ− (3.124) (3.125) denotes the value of the intermediate state, the state in the region between the fastest right and fastest left wave. Implementing this algorithmically, we define, λ+ = max(0, λR , λL ), (3.126) λ− = min(0, λR , λL ), (3.127) where λ+ is a right-going wave characteristic, and λ− is a left-going wave characteristic. λL are the eigenvalues of A(qL ) the Jacobian evaluated using the state values from the left reconstruction. λL are the eigenvalues of A(qR ) evaluated using the state values qR determined from the rightreconstruction. The 0’s are included to allow for the case where there are either no right-going or left-going waves, i.e. in the instance that all characteristic velocities are travelling in the same direction. Using these characteristics we reconstruct the flux [143], FHLL = λ+ f (qL ) − λ− f (qR ) + λ+ λ− (qR − qL ) . λ+ + λ− (3.128) This method is known to be very diffusive, but has proven to handle extreme shocks very well (e.g. pressures may vary between neighbouring cells by several orders of magnitude). This method is well suited for our surveys. 91 3.8. THE GODUNOV METHOD Lax–Friedrichs The final flux calculation we used was the classic Lax–Friedrichs (LF) method, which may be interpreted as an integral average within a cell [20] and has the form, q∗ = 1 ∆t (qL + qR ) − [f (qR ) − f (qL )] . 2 2∆x (3.129) a 1 [f (qL ) + f (qR )] − (qR − qL ), 2 2 (3.130) We then calculate the flux to be F= where the value a may be determined by, a = max |λp | p ∆x , ∆t (3.131) where λp are the local eigenvalues for the system. This reduces to the simple LF method when the maximum eigenvalues tend to unity. This equation has the form of an average of the two fluxes at an interface with an additional term in the equation that takes on the form of a numerical diffusion to damp out instabilities [19]. This method is very diffusive and “smears” badly, only giving usable results for very fine meshes. In practice, this method is used when the eigenvalues are approximately unity [123]. The final form of the flux approximation is; F= f (qL ) + f (qR ) + a (qR − qL ) . 2 (3.132) This is the most dissipative shock-capturing method of those considered in this study, and is best used for code development. Before we proceed to the discussion of the remaining parts of the solution algorithm we discuss limitations of the different flux approximations. 3.8.4 Limitations of Approximate Riemann Solvers The methods listed above are approximations to the full Riemann problem. These are used for maximum efficiency, as the full Riemann solver is numerically expensive and takes too long to 92 3.8. THE GODUNOV METHOD perform parameter surveys for different fluid configurations. We have mentioned that each of the methods have different strengths and weaknesses. For the study at hand we use the Roe solver and the HLL solver, as these methods produced the best results. The other solvers were implemented for their simplicity and for code testing purposes only, as they were too diffusive. Quirk [144, 36] discuss limitations of each of the techniques, and found that there were two modes of failure when using the Roe method: first, that it allowed for entropy violations; and second, that near vacuum it had difficulty preventing negative internal energy/negative pressure from forming. There have been proposed entropy fixes that will ensure that the internal energy will be positive. He does warn that these procedures are diffusive and tend to be no better than the artificial viscosity methods, in which we change the fundamental equations of motion by adding dissipative terms in order to retain the use of finite difference techniques. Quirk went on to say that the HLL methods were the best for shock handling while still maintaining positive pressures. Our experience has been that it will depend on what physical properties we are trying to study as to which method needs to be used. By implementing enough tests for solutions that violate physics in the code, the entropy violations and negative internal energies could be caught, preventing the production of unphysical results. 3.8.5 Basic Algorithm The algorithm used to perform the two dimensional evolution goes as follows: 1. Calculate the state value of the cell at the interfaces along one spatial dimension, say x1 from cell values Qn ; 2. Calculate the flux integrals in Eqn. (3.96) using an approximate Riemann solver for fluxes along the x1 -direction; 3. Repeat step 1 along a second spatial dimension, say x2 ; 4. Calculate the flux integrals in Eqn. (3.96) using an approximate Riemann solver for fluxes along the x2 -direction; 5. Calculate the source integral in Eqn. (3.97); 6. Perform the time step to determine Qn+1 using (3.99). 93 3.8. THE GODUNOV METHOD xi+1/2 ∆t λ Qni−1 Qni Qni+1 Figure 3.11: The CFL condition for a one dimensional cell. For the solution of the conservation equations (3.1) to be convergent we require that the time step, ∆t, be small enough that the characteristics do not have enough time to interact with cells beyond the adjacent cell. This condition must be met for convergence to be possible; however, satisfying this condition alone does not guarantee convergence. These are the steps we follow when solving our multidimensional magnetohydrodynamic system of equations. The actual time integration is performed using explicit ODE integration techniques discussed in Sec. 3.8.7. Before we discuss the conditions and methods that we use to perform the time integration. We briefly introduce a condition, known as the CFL condition, required for the stability of the time integration of the numerical solver. 3.8.6 The Courant–Friedrichs–Lewy (CFL) Condition The CFL condition is a requirement originally posed for finite closed form mathematics [145]21 to prove the existence of solutions for different PDEs. They prove that a sequence of approximate solutions, as a smaller and smaller grid is used, converges to the continuum solution. Since the original concept, numericists have used this as a necessary condition for their approximate solutions to be convergent to the true solution. If we consider a solution of the general system Eq. (3.1) at some coordinate in space and time, say Qni , it depends on the data at different spatial coordinates in a previous timestep. In particular, if we consider the variable reconstruction techniques used, we know any flux value will depend on at least three cells. The characteristic values, λ as described in Eqns. (3.48, 3.49, 3.53), in the centre cell (Fig. 3.11) give the wave speeds and any information carried by the waves will propagate a distance λ∆t. If the time step is too large, the waves from the centre cell will propagate beyond 21 The original paper is written in Russian [146] 94 3.8. THE GODUNOV METHOD the adjacent cells [19]. The CFL or Courant factor, ν, for a (magneto)hydrodynamic system is defined to be, ν= ∆t max |λp |, ∆x p (3.133) where p is the index that runs over all possible characteristics for the GRMHD system. For flows of higher dimension, the overall maximum characteristic from the Jacobians of all dimensions is used; for example, in two spatial dimensions, ν= 1 2 ∆t max |λxp |, |λxp | , ∆x p (3.134) i where λxp denotes the set of eigenvalues from the ∂f i /∂q Jacobian. From this criterion we know the CFL condition is governed by the largest overall characteristic wave speed. To perform the explicit time integration we use the method of lines, as described in the following subsection. 3.8.7 Method of Lines We calculate the time integration by using an explicit method where we split the temporal and spatial differencing. Using the method of lines, we re-write the partial differential conservation equations as a set of first order ordinary differential equations, dq = L(q), dt (3.135) where L contains all spatial differential operations. Based on our flux approximations the exact form for this ODE is, dQi = Fi+1/2 − Fi−1/2 + s(Qi , gµν ), dt (3.136) where, as before, Qi is the cell average of the i-th cell, Fi+1/2 , and Fi−1/2 are the numerical fluxes as described in Sec.3.8.1, and s is the source term evaluated at Qi . Now that the flux form is written as an ODE, we use standard solution techniques for ODEs. Below we outline two methods, Euler and Runge–Kutta, used to solve the equations of motion [19]. 95 3.9. BOUNDARIES Euler This is a first order accurate method. For an ordinary differential equation of the form y ′ = f (t, x(t)), the algorithm is: Qn+1 = Qni + ∆t Fi+1/2 − Fi−1/2 + S(Qni , gµν ) . i (3.137) This method is explicit22 , with the local error proportional to ∆t2 . We also see that it comes from a simple finite difference approximation to the original ODE. Runge–Kutta The Runge–Kutta method is outlined below: (1) = Qni + ∆t Fi+1/2 (Qn ) − Fi−1/2 (Qn ) + S(Qni , gµν ) ; (2) = Qi Qi Qi Qn+1 i (1) (1) + ∆t Fi+1/2 (Q(1) ) − Fi−1/2 (Q(1) ) + S(Qi , gµν ) ; 1 (1) (2) = Qni + (Qi + Qi ). 2 (3.138) (3.139) (3.140) This method uses a trial time step evaluated at the midpoint of each time step. Using this information, it corrects the final solution of each time step. 3.9 Boundaries The finite volume method as described above is defined to work inside the body of a domain. The boundaries of the domain must be treated as special cases. We use extra cells around the domain, commonly referred to as ghost cells, and update these cells based on the physical setup of the problem at hand. The ghost cells are not part of the physical domain, and are introduced for algorithmic simplicity. For the following, q1 and qN denote the quantities in the cells at the edge of the domain, while q0 and qN +1 denote the values of the quantities in the ghost cells. Typically, the boundary updates will be some combination of the following cases: 22 Explicit methods refer to methods that calculate the future solution to differential equations using the information available at the current time step. This is in contrast to implicit methods that calculate the future solution to differential equations using both future and current time information. We refer the reader to [19, 20] for more details. 96 3.9. BOUNDARIES • Periodic boundary conditions (BCs): q0 = qN ; (3.141) qN +1 = q1 . (3.142) – these are not really boundaries, but for computational implementation we simply match the values of the ghost cells to the values of opposite side of the domain. • Simple Outflow BCs: qN +1 = qN ; (3.143) q0 = q1 . (3.144) – these are used when a wave is expected to leave the domain and not return. Once out of the domain, the wave has no other impact on the system, thus the name “outflow”. We use zeroth order extrapolation to calculate the value of the ghost cells [19]. • Extrapolated outflow BCs: q0 qN +1 = 2q1 − q2 ; (3.145) = 2qN − qN −1 . (3.146) – these are used when a wave is expected to leave the domain and not return, this modification prevents waves from artificially reflecting from of the outflow boundary. In this case we use a first order extrapolation to determine the value of the ghost cells [19]. • Inflow BCs qN +1 = const. (3.147) – here the boundary cells are held constant through the entire simulation. This boundary is also known as a “prescribed” boundary condition. 97 3.9. BOUNDARIES • Reflective (Wall) [36]: v⊥ (xWall ) = 0; (3.148) B⊥ (xWall ) = 0; (3.149) v⊥ (xWall + ∆x) = −v⊥ (xWall − ∆x); (3.150) B⊥ (xWall + ∆x) = −B⊥ (xWall − ∆x); (3.151) = q(xWall − ∆x). (3.152) q(xWall + ∆x) – here a vector component perpendicular to the boundary say, v⊥ , is zero at the “wall”. Any ghost cell beyond the “wall” will take on the negative of the value of a cell adjacent to the wall. In essence this treatment will look as though two fluid elements are headed towards each other near the wall. We take the wall to be located at x = xWall . For simplicity, we consider functional dependence on one spatial coordinate only. All other quantities are reflected, including the components of the velocity that are parallel to the wall. Note that the magnetic field components are treated in the same way as the velocity field components. The boundary conditions presented above are sufficient for all physical setups used in this thesis, including the Bondi–Hoyle accretion problem that will be discussed in Chap. 5. 3.9.1 The Floor For relativistic fluids, one of the possible outcomes of the physical system could be two adjacent cells with dominant velocity components that face away from each other. This occurrence will produce a physical vacuum, a state in which the density goes to a value near zero. On occasion the cell interface reconstruction will produce a negative pressure, likewise, the primitive variable solver as discussed in section 3.3, is known to experience difficulties finding the correct root. The result is typically an unphysical solution, such as a state with negative pressure. When the unphysical states occur we check the value of the energy, compared to the baryon density. To circumvent this issue we use a numerical floor to impose a lower bound on the pressure and density. This floor is typically imposed on the conservative variables before calculating the primitives. The concept of a floor is discussed in many different physical setups, and there does not appear to be a general solution that works for all cases. In our implementation, we found it fruitful to fix 98 3.9. BOUNDARIES the floor to some constant low value, regardless of the coordinate location, such as those discussed in Noble et al. [125]. The method of floor decided upon for this project, after a discussion with Nielsen [147], was to determine a maximum between a floor value and the conservative density, and at the same time between the floor and the energy, τ : D= D, if D > floor; (3.153) floor, otherwise; τ, if τ > floor; τ= floor, otherwise. (3.154) After the primitive variable recovery has been performed, we perform a second set of tests on the primitive density and pressure: ρ0 , if ρ0 > floor; ρ0 = floor, otherwise; P, if P > floor; P = floor, otherwise. (3.155) (3.156) In the case when the primitive variables were altered, the conservative variables are recalculated to ensure self consistency. It is important to understand that the floor is not a physical quantity, but rather a numerical control. It has been observed in our system that this quantity is only necessary in ultrarelativistic systems, and does not interfere with any measurements, since any floored regions only occur in the downstream region and are found to propagate downstream out of the domain. In our study we found that the floor is only necessary for the ultrarelativistic flows with speeds close to the speed of light. In this chapter we have explained the characteristics and hyperbolicity conditions of a system of partial differential equations. We explained the algorithm and techniques used to calculate each step and with them we are able to calculate the evolution of the fluid equations of motion. Finally we explored limitations of our implementation of this algorithm. We now describe the methods we 99 3.9. BOUNDARIES used to test the code developed, and present the results. 100 CHAPTER 4 NUMERICAL ANALYSIS AND TESTS The purpose of numerical code testing is to ensure that the numerical results reflect a true solution of the equations being solved. We use three approaches to this, the first of which is convergence testing, where we run the simulation at a minimum of three different resolutions to determine if the solution is showing a trend to the continuum solution. The second is an independent residual test where we take the solution as determined by a numerical method and substitute it into an alternative numerical approximation. As the solution approaches the continuum solution, the independent residual is expected to tend to zero. The previous two tests are general numerical tests which are applied to determine if the numerical solution tends toward the continuum solution. The last set of tests of our solver are specific to the finite volume method, and test the shock-capturing properties of the code by looking at specific situations where the solutions are known. There are a battery of tests that may be applied for verification of the shock capturing properties [47, 52, 55]; in particular, we consider one dimensional shock tube tests and blast waves. We also consider tests in two spatial dimensions including the hydrodynamic double shock wave, or Riemann test, and the magnetized strong blast wave. As a stronger comparison between the different flux approximations we use, we also investigate the Kelvin–Helmholtz instability [142]. Finally, we consider the relativistic Bondi accretion problem as a steady state problem, used to test our ability to handle time evolution of a steady system in a general relativistic context. 4.1 Convergence The following are based on notes written by Choptuik [148], as well as the paper [149]. We can write a differential equation in a general form, L[u] = f, (4.1) 101 4.1. CONVERGENCE where L represents the differential operators. This representation hides all dependence of the equations on the variables and parameters in the differential equation. u = u(t, xi ) represents a solution vector and f = f (t, u, xi ) is a source term. If we let Lh be a discretized differential operator for some grid spacing h, we further write the discretized equation, Lh [uh ] = f h , (4.2) where uh = uh (t, xi ) and f h = f h (t, uh , xi ) are the discretized solution and source vectors, respectively. The continuum solution, u even evaluated at discrete points, is not a solution of (4.2). The degree to which the continuum solution fails to satisfy the discrete equation defines the truncation error, τ h , τ h = Lh [u] − f h . (4.3) This error may be found from the Taylor series expansion of the difference operators. Since we are actually interested in the solution u, we require that our approximate solution tends to the true solution uh → u as the resolution increases. We further define a solution error, eh = eh (t, xi ), containing all errors that arise from the substitution of the continuous problem with the discrete problem, eh = u − uh . (4.4) One error that is present in all finite precision numerical analysis is round-off error. This error can play a large role in fluid evolution due to the strongly coupled nature of the equations, and the fact that each field can vary over several orders of magnitude relative to each other. The wide range of magnitudes indicates that rounding errors may quickly overwhelm a numerical solution. Using the Richardson expansion23 [148, 149], we are able to make a connection between the solution error order and the order of the truncation error, resulting in; uh = u + h2 e2 + h4 e4 + O(h6 ). (4.5) To determine the convergence of a differencing scheme to the continuum problem, we must test Richardson expansion is based on the Richardson ansatz wherein, if any solution, uh , of any finite difference approximation which uses a uniform mesh structure with scale parameter h, and is completely centred should, in the limit h → 0, have the expansion given in the text [148]. 23 The 102 4.1. CONVERGENCE the results at different resolutions and determine the convergence of the solution uh to the true solution u, without knowing what it is. To proceed with the discussion of our convergence we need to introduce two additional concepts, a vector or array norm, as well as a convergence factor. 4.1.1 Norms When investigating convergence, we require techniques that allow us to represent arrays of data as single data points. Typically this is done by measuring a norm of a data set. In numerical analysis, when we refer to a norm, we consider a p-norm. Say we have a vector solution uh of a discrete equation such as in equation (4.1): Lp (uh ) = 1 N 1 p N i=i |uhi |p ≡ uh p, (4.6) where N denotes the number of elements in the solution vector. To extend this to higher dimensions, we use Lp (uh ) = 1 N1 N2 N3 N1 ,N2 ,N3 i=i,j,k p1 |uhijk |p ≡ uh p , (4.7) where Ni denotes the number of elements of the solution array uijk in the ith direction. In this study, unless otherwise specified, when we discuss the norm of a solution we mean the L2 -norm i.e. p = 2 and drop the subscript. 4.1.2 Convergence Factor If at a minimum we have three different resolutions h1 < h2 < h3 , then we evaluate the Richardson extrapolation, Eqn. (4.5) as: u h1 = u + h21 e2 + h41 e4 + O(h6 ); u h2 = u + h22 e2 + h42 e4 + O(h6 ); u h3 = u + h23 e2 + h43 e4 + O(h6 ). (4.8) When we discuss the order of a polynomial expansion in this context we refer to the powers on the expansion parameter, h. If the power on the expansion parameter, h is one, the system is said to 103 4.2. INDEPENDENT RESIDUAL be first order. The equations (4.8) are second order. We then define a convergence factor, Q, to be Q≡ u h3 − u h2 . u h2 − u h1 (4.9) With the expansions (4.8) and keeping only lowest orders in h, this reduces to Q= h23 e2 − h22 e2 h23 − h22 (h3 /h2 )2 − 1 = = . h22 e2 − h21 e2 h22 − h21 1 − (h1 /h2 )2 (4.10) Typical analysis of this type uses grids that are scaled by factors of two, i.e. h1 = h, h2 = 2h, and h3 = 4h. In this case we expect that the convergence factor Q = 4. Using the above method does not produce an integer convergence factor but does allow us to study the convergence properties of the numerical routine. For the systems that we study, we extend this analysis to two and three dimensions. Since we keep the relative sizes of the mesh uniform, i.e. when we increase the resolution in one dimension we likewise increase resolution in all other dimensions, we are free to choose the lowest resolved dimension of the system to determine the error in our system. Due to the variable reconstruction method discussed in Chap. 3, we expect that the solution of the system of equations to reduce to first order accuracy near discontinuities and extrema. 4.2 Independent Residual To ensure that the solution of the given set of equations is the correct solution, we can also use an independent residual. To use this method we solve the equations of motion using one method of discretization, then discretize the same equations with a second independent technique, and look at the error left when applying the second method to the solution from the first. The error is referred to as the residual. We check the rate of convergence of the residual to zero, which depends on the methods used. For example If we use a second order solver to obtain the solution, and check our solution using a first order independent technique the rate of convergence of the independent residual will be first order. If the solutions do not converge, then we know there is a problem with the implementation. In ¯ h which is different from the original short, we consider a linear discretized differential operator L 104 4.2. INDEPENDENT RESIDUAL discretization Lh . We then calculate the residual, rh , rh = L¯h (uh ) − f h , (4.11) where uh is the solution of the system Lh (uh ) = f h . If Lh and L¯h are the same continuum operator, then the residual will converge as expected with the Richardson extrapolation [36]. In fluid dynamics, the equations of motion are highly nonlinear. In the following we will switch notation for clarity. Since we are interested in two dimensional problems which may not use equal grid spacing, for clarity we use ∆x and ∆y in lieu of h to represent the grid spacing in different dimensions. We further denote temporal grid spacing using ∆t. We use the following first-order finite-difference discretization to evaluate the independent residual; r= n+1 1 2 (qi+1 − qni+1 ) + 12 (qn+1 − qni ) i + ∆t n fi+1 − fin ∆x − Sni+1 + Sni , 2 (4.12) where r are the independent residuals for each of the corresponding state variables q. Recall that n fi,j = f (qni,j ) and Sni,j = S(qni,j ). For two dimensional systems, the above independent residual is; r = n+1 1 4 (qi+1,j + n+1 n+1 1 1 n n n − qni+1,j ) + 41 (qn+1 i,j − qi,j ) + 4 (qi,j+1 − qi,j+1 ) + 4 (qi+1,j+1 − qi+1,j+1 ) ∆t n n n n Sni+1,j + 2Sni,j + Sni,j+1 fi+1,j − fi,j fi,j+1 − fi,j + − , (4.13) ∆x ∆y 4 which is also first order. To evaluate the independent residual above we use the cell-centred values for each variable. We used the algebraic form of the fluxes found in Eqns. (2.118) and (2.123) to calculate the finite differenced fluxes for the magnetohydrodynamic and ultrarelativistic systems respectively. Because the independent residual is only a first order approximation to the partial differential equations, we can only expect our test to converge to zero to first order. We expect some difficulty using this method to test global convergence because this difference scheme is not defined across discontinuities. Although we only be use this method to verify the solution array, not solve for it, we will still have poor resolution of any discontinuous variable in the domain. Recall that using a finite difference approximation to solve for solution across a discontinuity leads to unphysical oscillations. Using a finite difference technique as outlined above to check for convergence of a solution that contains shocks may lead to misleading error estimates when calculating global errors such as those calculated using Eqn. (4.9). We applied the 105 4.3. SHOCK AND SYMMETRY CAPABILITIES independent residual test for systems that had smooth solutions and terminated the test before the shocks form. To determine how well our numerical scheme could handle shocks, we used different shock tests which are described in the next several sections. 4.3 Shock and Symmetry Capabilities Due to the number of different Riemann approximation schemes as well as magnetic field constraint handlers, we need a set of problems that test the shock-capturing properties of, and constraint violations of the schemes in fluid systems. The first set of tests we describe are one dimensional and are performed using Minkowski spacetime in Cartesian coordinates. They are designed to test the different flux approximations for their ability to preserve the sharp discontinuities experienced at a shock interface. These tests are commonly referred to as the shock tube tests and are tests of the hydrodynamic codes. Balsara [47] extended these tests to include magnetic field contributions. For the following sections, unless stated otherwise, we use Cartesian coordinates and restrict our attention to the spatial domain x = [−0.5, 0.5]. 4.3.1 Sod Shock Tube Tests The Sod shock tube [19] is one of the original systems investigated using the finite volume method in one dimensional hydrodynamics. A shock tube contains a removable membrane that separates two regions, denoted left, L and right R, where the fluid state variables, p(0, x) of the two halves are of different pressure and density. Both fluids are initially at rest. We choose the location of the membrane to be the origin, x = 0. We initialize the primitive variables: (ρ0 (0, x), v x (0, x), P (0, x)) = (ρ0L , 0, PL ), (ρ0R , 0, PR ), if x < 0; (4.14) if x ≥ 0. Here ρ0L , ρ0R , PL , PR are all constant. By judicial choices of (ρ0L , PL ) and (ρ0R , PR ) we set the speed of sound to be the same for both sides of the discontinuity. This model tests the shock capturing properties of a hydrodynamic code since the solution contains three different types of waves: a shock wave; a rarefaction wave; and a contact discontinuity. Results of this test may be seen in Fig. 4.1. 106 4.3. SHOCK AND SYMMETRY CAPABILITIES Figure 4.1: The Sod tube test [19]. Plotted are the pressure, density, v x (middle), and Sx (bottom). The interesting parts of this plot include the existence of a rarefaction wave, a shock wave, and a contact discontinuity where the pressure is constant but the density exhibits a sharp discontinuity. This is a result of using the HLL solver with ch = 1.0, and Nx = 6400. The solution shown here is at t = 0.4, initial data are in the first two rows of table 4.1. 107 4.3. SHOCK AND SYMMETRY CAPABILITIES Test Sod Balsara 1D Blast Wave L R L R L R ρ 1.0 0.125 1.0 0.125 1.0 1.0 P 1.0 0.1 1.0 0.1 1000.0 0.1 vx 0.0 0.0 0.0 0.0 0.0 0.0 vy 0.0 0.0 0.0 0.0 0.0 0.0 vz 0.0 0.0 0.0 0.0 0.0 0.0 Bx 0.0 0.0 0.5 0.5 2.0 2.0 By 0.0 0.0 1.0 -1.0 7.0 0.7 Bz 0.0 0.0 0.0 0.0 7.0 0.7 Table 4.1: Initial conditions for the 1D relativistic Balsara blast wave, and the 1D strong blast wave [47]. Since we want the code to handle magnetic field contributions as well, we focus on further tests which probe the the magnetic field contributions. These are the Balsara blast wave [47] and the strong shock blast wave which are based on the hydrodynamic Sod Shock tube as described above. 4.3.2 Balsara Blast Wave In this test, magnetic fields are introduced to a Sod shock tube by using a constant Bx component and a discontinuous By component. In the Balsara blast wave, the shock-capturing properties of a magnetohydrodynamic code are tested, as the solution contains all of the relevant types of waves, including coupled magnetosonic waves. The initial data for a test in the x ˆ-direction are: (ρ0 (x, 0), v x (x, 0), P (x, 0), B x (x, 0), B y (x, 0)) = (ρ0L , 0, PL , B x (x, 0), B y ), L if x < 0; y (ρ0R , 0, PR , B x (x, 0), BR ), if x ≥ 0. (4.15) y Here ρ0L , ρ0R , PL , PR , B x (x, 0), BLy (x, 0), BR (x, 0) are all constant. The magnetic field, B x is uni- form over the entire domain and constant in time. The quantities used to initialize the variables can be found in table 4.1. The results of the Balsara blast wave at time t = 0.4 for the initial data in 4.1 are seen in Fig. 4.2. They are in agreement with solutions found in the literature [122, 47]. This test was applied in both spatial directions as a verification that the flux approximations used are valid. 4.3.3 Magnetized Strong Blast Wave In an extension to the Balsara blast wave, we test the numerical solver’s ability to capture strong shocks. This one dimensional test is configured as seen in table 4.1. Using the HLL solver with both the MC and minmod limiter, we are able to capture the strong shocks. The HLL method is a 108 4.3. SHOCK AND SYMMETRY CAPABILITIES Figure 4.2: The Balsara blast wave test as designed by Balsara [47]. Plotted are the pressure, density (top), v x (bottom left), B y (top right), and v y (bottom right). The interesting parts of this plot include the existence of a shock wave, a rarefaction wave, Alfv´en wave and a contact discontinuity where the pressure is constant but the density exhibits a sharp discontinuity. This is a result of using the HLL solver with ch = 1.0, and Nx = 6400. The solution shown here is at t = 0.4, initial data are in the second two rows of table 4.1. 109 4.3. SHOCK AND SYMMETRY CAPABILITIES Figure 4.3: The density plot of the relativistic Balsara blast wave test [47]. When compared to Fig. 4.2, we label the different wave forms that develop. It is noted that some of the wave combinations are unique to magnetohydrodynamic systems. The labels are as follows: LR denotes a left moving rarefaction wave; and CW denotes a compound wave. The compound wave is unique to MHD systems and consists of an over-compressive shock, and a rarefaction wave. CD denotes a contact discontinuity, which is easily recognized by noticing that the pressure is constant across this discontinuity as seen in Fig. 4.2. SS denotes a slow shock and RR denotes a right moving rarefaction [150, 151]. 110 4.3. SHOCK AND SYMMETRY CAPABILITIES Figure 4.4: The Convergence properties of the Balsara blast wave test. The green line uses Nx = 1600, red denotes Nx = 3200, and black denotes Nx = 6400. We see that the discontinuities are becoming sharper as the resolution increases. The regions −0.01 ≤ x ≤ 0.15, and 0.1 ≤ x ≤ 0.16 are shown with greater magnification in Fig. 4.5. 111 4.3. SHOCK AND SYMMETRY CAPABILITIES Figure 4.5: The convergence properties of the Balsara blast wave. Level 1 uses Nx = 1600, level 2 uses Nx = 3200, and black denotes Nx = 6400. Here we focus our attention on the regions −0.01 ≤ x ≤ 0.15 (top) and 0.1 ≤ x ≤ 0.16 (bottom). As the resolution increases we are able to better approximate the sharp discontinuities and extrema in the profile. 112 4.3. SHOCK AND SYMMETRY CAPABILITIES diffusive flux approximation method. The diffusion is shown near strong shocks in particular when we use the minmod limiter. When we use the HLL flux approximation with the MC limiter we see less diffusion in the same region. This is clearly displayed in Figs. 4.6 and 4.7. Figure 4.6: The one dimensional strong blast wave. We use the domain −0.5 ≤ x ≤ 0.5, with a resolution Nx = 401. The dotted line represents the results while using of the MC limiter, while the dashed line represent the result of using the minmod limiter. The exact solution is represented by the solid line [52]. This shows evidence that the MC limiter offers a better representation of shocks. This is evident, not only due to the larger density amplitude, but also in the region around x = 0.4 as seen in the inset plot. 4.3.4 Two Dimensional Riemann Tests We present the case where shocks are present in multiple directions. Unfortunately, a simple double shock system, such as a two dimensional Balsara type setup, does not exist for the magnetic field case, due to the necessity of the no-divergence constraint in the initial conditions. In Fig. 4.8 we 113 4.3. SHOCK AND SYMMETRY CAPABILITIES Figure 4.7: The one dimensional strong blast wave at a higher resolution. We use the domain −0.5 ≤ x ≤ 0.5, with a resolution Nx = 1601. The dotted line represents the results while using of the MC limiter, while the dashed line is the result of using the minmod limiter. The exact solution is the solid line. The higher resolution test shows evidence that the MC limiter offers a better representation of shocks than the midmod limiter. Once again the density amplitude while using the MC limiter is closer to the exact solution [52], but also in the blown up region around x = 0.4 we see that the MC limiter begins to capture the contact discontinuity that the minmod limiter is too diffusive to capture. 114 4.4. KELVIN HELMHOLTZ INSTABILITY Test 2D Riemann Shock Test Location BL BR TL TR ρ 0.5 0.1 0.1 0.1 P 1.0 0.01 1.0 0.01 vx 0.0 0.0 0.99 0.0 vy 0.0 0.99 0.0 0.0 Table 4.2: Initial conditions for the 2D relativistic Riemann shock wave. BL = Bottom-Left, TL = Top-Left, BR = Bottom-Right, and TR = Top-Right. These initial conditions are provided by FLASH, http://flash.uchicago.edu/website/codesupport/flash3_ug_3p2/node31.html. present the result of the 2D shock test for a purely hydrodynamic system (i.e. Bx = By = 0), at time t = 0.7. We restrict our attention to Cartesian coordinates, with domain, −0.5 ≤ x ≤ 0.5, −0.5 ≤ y ≤ 0.5. 4.4 Kelvin Helmholtz Instability Fluids may become unstable when the gradient of any of the constituent fields, (ρ0 , v i , P, B j ), exceed certain thresholds. As is discussed in Biskamp [152], when the convective transport of the field flux is more efficient than the diffusive transport due to viscous effects, this threshold is reached. These instabilities are categorized as “Kelvin–Helmholtz”, “Rayleigh–Taylor”, and the “tearing mode” instability. It is also known that the presence of magnetic fields may stabilize these instabilities. Since we consider ideal fluids, the fluid does not experience viscous effects which leads the way for any of the previously stated instabilities. The Kelvin–Helmholtz instability (KHI) occurs when there is a large enough gradient of the state variables across a shear flow within a fluid. If we perform a Fourier analysis of the momentum equations of motion, the dispersion relation reveals a growing mode proportional to the gradient across the velocity field. Further analysis indicates that the presence of a sufficiently strong parallel magnetic field is expected to stabilize the system. Astrophysical fluids may experience differential rotation, which corresponds to a shear flow. When performing this investigation we used the same setup as described within the ATHENA website24 . Since this is not the focus of this thesis, we will only go into implementation details, for a full treatment of this phenomenon please see [142, 152, 153, 154, 155]. The initial conditions are reproduced in Table 4.3. We use periodic boundary conditions in 24 http://www.astro.princeton.edu/∼jstone/tests/index.html 115 4.4. KELVIN HELMHOLTZ INSTABILITY Figure 4.8: The two dimensional Riemann shock test. We present the baryon rest mass density, ρ0 (left) and the pressure, P (right). The top plot shows the initial conditions, while the bottom plot shows the result of an evolution at t = 0.7. We used a resolution of 1025 × 1025. The initial conditions are seen in Table 4.2. This shows the codes capability for capturing shocks while also maintaining symmetry. Even with using a second order scheme, we are able to capture a lot of the features of this system including the jet that protrudes out the back of the shock. The domain is (x, y) = [−1, 1] × [−1, 1]. 116 4.4. KELVIN HELMHOLTZ INSTABILITY Weak Strong ρ 1.0 2.0 1.0 2.0 L R L R P 2.5 2.5 2.5 2.5 vx 0.5 -0.5 0.5 -0.5 vy 0.0 0.0 0.0 0.0 Bx 0.5 0.5 5.0 5.0 By 0.0 0.0 0.0 0.0 βp 20 20 0.2 0.2 Table 4.3: Initial conditions for the 2D Kelvin–Helmholtz tests. When the magnetic field is not considered, B x is set to zero. The domain, (x, y) = [−0.5, 0.5] × [−0.5, 0.5], was broken into three regions for −0.5 ≤ y < −0.25 and 0.25 < y ≤ 0.5 we used the parameters labelled L, for −0.25 ≤ y ≤ 0.25 we used the parameters labelled R. both the x and y direction. All the state variables: (ρ0 , v x , v y , P, B x , B y ) = (ρ0 (t, x, y), v x (t, x, y), v y (t, x, y), P (t, x, y), B x (t, x, y), B y (t, x, y)) (4.16) are now functions of two spatial dimensions and time. The setup at t = 0 is as follows: (ρ0 , v x , v y , P, B x , B y ) = (ρ0L , v x , P, B x , B y ), L L if −0.25 < y < 0.25; y x (ρ0R , vR , P, B x , BR ), otherwise. (4.17) A uniform pressure, P , and magnetic field, B x , cover the entire domain. In the centre of the domain the fluid velocity, v x , and density are discontinuous in the yˆ-direction. The vertical velocity, v y , is initially zero. We refer the reader to Table 4.3 for the exact values used. For all velocity fields, we use a periodic perturbation to trigger the instability; the amplitude of the velocity perturbation, vp , is set to 1% of the original field strength. The velocity components are vx = v0x + vp cos(2πyl) cos(2πxl), (4.18) vy = vp cos(2πyl) cos(2πxl), (4.19) where l is used to control the frequency, and v0x represents the original uniform velocities vLx and x vR . The instability that develops is known as the Kelvin–Helmholtz instability (KHI). For the study of non-relativistic form of this instability, we refer the reader to Frank et al.[142]. The study of this set-up is relevant when testing the numerical diffusion that occurs when using different flux approximations. The only flux approximation that we developed that captures this instability is the Roe solver. All other flux approximations are too diffusive and damp out the instability. When the magnetic field is embedded in the fluid system, the flow is more stable and the strong 117 4.4. KELVIN HELMHOLTZ INSTABILITY Figure 4.9: Baryon density evolution results of the Kelvin–Helmholtz initial conditions in a special relativistic system. 512 × 512 resolution using a periodic velocity field perturbation. From top to bottom, T = 5, 10, 15. The formation of the “Cat’s Eye”, the name for the vortices resulting from the unstable flow, is apparent. 118 4.4. KELVIN HELMHOLTZ INSTABILITY Figure 4.10: Baryon density evolution results of the Kelvin–Helmholtz initial conditions in a special relativistic system. 1024 × 1024 resolution using a periodic velocity field perturbation. When we compare this to the 512 × 512 simulation there is a noticeable increase in detail as well as an increase in the number of vortices. This is a feature of unstable systems, the larger the resolution the greater the number of features, which might lead us to believe the system is not convergent. Higher resolution will result in capturing even more vortices. 119 4.4. KELVIN HELMHOLTZ INSTABILITY Figure 4.11: Baryon density evolution results of the magnetized Kelvin–Helmholtz initial conditions (table 4.3) in a special relativistic system. 512×512 resolution using a sinusoidal velocity field perturbation to prevent ∇ · B = 0 violations in the initial steps. From top to bottom T = 5, 10, 15, left column uses B x = 0.5, the right column uses B x = 5.0. Note that the presence of a relatively strong magnetic field suppresses the instability entirely, offering an example where the presence of a magnetic field stabilizes a flow. The initial plasma beta for the left column is βp ∼ 20, while on the right βp ∼ 0.2. 120 4.5. RIGID ROTOR vortices seen in the hydrodynamic system are reduced. If the magnetic field is weak, as in the left column of Fig. 4.11, the instability persists, but takes longer to fully develop. If the magnetic field is sufficiently strong in the direction of the fluid flow, as seen in the right column of Fig. 4.11, the instability is suppressed entirely. The result is a smoother flow as the instability is suppressed by the embedded magnetic field. In the case where the embedded magnetic field is in a direction orthogonal to the fluid flow, the instability is completely suppressed. 4.5 Rigid Rotor The rigid rotor is a two dimensional special relativistic MHD test which evolves an initially rigidly rotating fluid in the presence of a magnetic field [123, 48]. We show the results using an HLL solver with a minmod limiter. The divergence constraint handling is monitored by measuring two different quantities, we consider the L2 -norm of both the divergence of the magnetic field, and the auxiliary field Ψ as seen in Fig. 4.12. The resulting density evolution may be seen in Fig. 4.13. To calculate the divergence of the magnetic field as described by (2.102), we use a spatially centred second-order finite-difference approximation. This discretization may be difficult to calculate across a shock and may consequently show strong violations as resolution increases (shocks become sharper). We used the second order finite difference approximation ∇·B → 1 √ γ + + √ x1 γB √ x2 γB √ x3 γB i+1,j,k − √ γB x1 2∆x1 √ x2 γB − i,j+1,k 2∆x2 √ x3 − γB i,j,k+1 2∆x3 i−1,j,k i,j−1,k i,j,k−1 . (4.20) In the smooth regions of the solution space, this approximation is second order accurate. The above approximation will break down when a shock occurs in the solution. We will need to consider another method to monitor the no-divergence constraint for the magnetic field, which will be discussed in section 4.6. The results of this test may be seen in the right image of Fig. 4.12. As we can see, discretizing ∇ · B = 0 using a finite difference stencil leads to misleading error estimates. In addition to the convergence of the magnetic field constraint, we also plot the convergence of 121 4.6. STEADY STATE ACCRETION the density cross section of the rigid rotor on both the x and y axis, as seen in Fig. 4.14. Figure 4.12: The convergence of the magnetic divergence constraint for the relativistic rigid rotor simulation. The base resolution is 321 × 321 (dotted line). It should be noted that the Lp -norms of the derived quantity ∇ · B = 0 may not be the most useful test for this particular constraint since the magnetic constraint violations are strongest near discontinuities. Other analysis techniques have been considered, but only the Lp norms appear to be universally accepted; therefore, these are the only techniques presented. The left shows the L2 norm of the deviation of auxiliary field Ψ from the expected continuum solution Ψ(x, y, 0), the right shows the L2 norm of the magnetic field constraint. If either of these fields blow up we know the evolution has become unphysical. In these evolutions, the quantities are bounded so the evolution variables may still be considered physical. These plots indicate that monitoring the constraint may be better performed by monitoring the auxiliary field Ψ, which is enforced by the fact if Ψ were to experience unbounded growth, the magnetic field constraint would experience strong violations. Due to the presence of shocks and extrema seen in this evolution, the convergence rate of this problem is less than second order. 4.6 Steady State Accretion When developing time dependent GRMHD codes, we use steady state problems as a test that the code and all methods used are functioning properly [28, 29, 55, 156]. The final two sections discuss the tests used to determine how well we can approximate a steady state solution using a time dependent code. Ultimately this tests the stability of the time evolution. 4.6.1 Spherical Relativistic Bondi Accretion The relativistic Bondi accretion problem consists of a spherically symmetric black hole accreting asymptotically stationary matter. This problem reduces to a simple ODE, and can be readily 122 4.6. STEADY STATE ACCRETION Figure 4.13: The results of the rigid rotor initial conditions with resolution 128 × 128, from top to bottom T = 0.4, 0.8, 1.2. The left column shows the results for the baryon density ρ0 , and the right column shows the results for the thermodynamic pressure P . The presence of a magnetic field is apparent in the first frame, where we see a flattened rotating fluid disk. 123 4.6. STEADY STATE ACCRETION Figure 4.14: The convergence of the relativistic rigid rotor simulation. The top shows the density cross section along the y-axis, the bottom shows the density cross section along the x-axis. The level 1 resolution (dotted line) is (Nx , Ny ) = 321×321, level 2 is (Nx , Ny ) = 641×641, and level 3 is (Nx , Ny ) = 1281 × 1281. We see that the base features of the evolution remain the same, becoming more pronounced and more sharply peaked. All cross-sections are recorded at time t = 0.4. 124 4.6. STEADY STATE ACCRETION Rigid Rotor ρ 10.0 1.0 I O P 1.0 1.0 vx −ωz y 0.0 vy ωz x 0.0 vz 0.0 0.0 Bx 1.0 1.0 By 0.0 0.0 Bz 0.0 0.0 Table 4.4: Initial conditions for the rigid rotor test. The I row denotes values inside a particular radius, while O denotes values outside of that radius. The rigid rotor is set to have zero velocity outside the radius 0.1, while inside the radius 0.1 the fluid is set to experience differential rotation defined by v = ω × r, where we use ω = (0, 0, 9.95). solved using standard numerical ODE solvers. This tests how well the time dependent code will solve a known time independent (steady state) problem. Here we work in spherical polar coordinates (t, r, θ) and assume that all state variables depend only on r. This reduces the PDE equations of motion to a set of ODEs: ∂r √ −g(Dˆ v r ) + ∂θ √ −g(Dˆ vθ ) = ∂r √ √ −g(Sr vˆr + P ) + ∂θ −g(Sr vˆθ ) = ∂r √ √ −g(Sθ vˆr ) + ∂θ −g(Sθ vˆθ + P ) = √ √ −g(τ vˆr + P v r ) + ∂θ −g(τ vˆθ + P v θ ) = ∂r 0; √ 1 µν −g T gµν,r ; 2 √ 1 µν −g T gµν,θ ; 2 √ −g αµ T µt − αΓt µν T µν . (4.21) (4.22) (4.23) (4.24) For the spherically symmetric black hole in either ingoing Eddington–Finkelstein or Schwarzschild √ coordinates, we have −g = r2 sin θ. We also assume that the fluid does not have any initial angular velocity, so v θ = 0, which leads to Sθ = 0, reducing our set of ODEs (4.21-4.24) to, ∂r √ −g(Dˆ vr ) = √ −g(Sr vˆr + P ) = √ −gP = √ −g(τ vˆr + P v r ) = ∂r ∂θ ∂r 0, √ 1 µν −g T gµν,r , 2 √ 1 µν −g T gµν,θ , 2 √ −g αµ T µt − αΓt µν T µν . (4.25) (4.26) (4.27) (4.28) Following Michel [25] and Hawley et al. [28] we define a temperature T = P/ρ0 , and integrate (4.25) to get 1 T Γ−1 ur r2 = C1 , (4.29) 125 4.6. STEADY STATE ACCRETION where ur is the radial component of the fluid 4-velocity. Then we integrate (4.26) to obtain 1 + (1 + 1 )T Γ−1 2 1− 2M + ur2 r = C2 . (4.30) This gives us enough information to solve for T (r) and ur (r) if we know the constants of integration C1 , C2 . To specify C1 , C2 we consider the transonic point, rc , the point through which the fluid speed transitions from subsonic (less than the speed of sound) to supersonic (faster than the speed of sound). With the information specified at the transonic point we are able to solve for the critical radial component of the 4-velocity, uc = uc (rc ) and the critical temperature Tc = T (rc ). For a detailed explanation, we refer the reader to [28, 29]. The steady state accretion test was performed in both one (r) and two (r, θ) dimensions. Although the phenomenon is one dimensional in nature, by performing the simulation with a free polar coordinate we will test the code’s capability of making angular approximations [55]. When we consider Eqn. (4.27) using Schwarzschild coordinates, the closed form solution reveals that the left side becomes √ ∂θ ( −gP ) = P r2 cos θ, (4.31) √ 1 µν −g T gµν,θ = P r2 cos θ. 2 (4.32) while the right side is Comparing (4.31) and (4.32) we see that (4.27) is satisfied trivially if calculated in closed form. In calculating the numerical solution to Eqns. (4.25–4.28) we use approximate solutions to the differentials as described in Chap. 3. The numerical solution of the left side of (4.31) is only approximately equal to the right side Eqn. (4.32). Solving (4.25–4.28) in two spatial dimensions reveals how well our techniques handle the approximate angular differentiation. If the code is developed properly, the divergence of the flux for the angular momentum term (4.31) will cancel the source term (4.32) within truncation. We show the evolution at time 100 and the L2 norm for the time derivative of the radial velocity to show the steady state behaviour. The results of this simulation may be seen in Fig. 4.16. For the simulation we used the Schwarzschild coordinates, and Γ = 5/3, rc = 8 to specify the problem. A final observation for the hydrodynamic code is that this initial condition proved to be a valuable test for the outflow boundary conditions. If we use the simple outflow boundary condition 126 4.6. STEADY STATE ACCRETION Figure 4.15: Convergence of the L2 -norm of the time derivative of the radial velocity for the relativistic spherical accretion, showing the convergence of the time derivative of the radial velocity. We note that despite the fact that the problem is fixed to a steady state initial condition, some numerical error occurs near the event horizon due to boundary conditions. After some transient time, the evolution maintains a steady state solution. The time derivative of the radial velocity is converging to zero. Only 5 ≤ t ≤ 15 is shown for emphasis. 127 4.6. STEADY STATE ACCRETION as described in Sec. 3.9, the evolution results in the radial velocity reflecting back from the event horizon due to the high pressure and density [57]. The extrapolated outflow boundary conditions prevent sharp unphysical reflections from occurring at this boundary. Figure 4.16: Relativistic spherical Bondi accretion. The top left is the radial velocity v r , the top right shows the thermodynamic pressure P , and the bottom shows the baryon rest mass density ρ0 . These results show the evolution at time ∼ 100. The exact solution is represented by the solid line. The only deviation from the exact solution occurs near the boundary; however, the simulation quickly settles to a steady state. 128 4.6. STEADY STATE ACCRETION 4.6.2 Magnetized Spherical Accretion We have seen that our numerical solution to the steady state spherical accretion problem is satisfactory, but this only tests the hydrodynamic solver. To test the full magnetohydrodynamic numerical solution against a steady state problem, we impose a radial magnetic field on the spherical accretion and obtain a steady solution [57]. First we ensure that the magnetic field will be divergence free. Thus we solve the ODE √ ∂r ( γB r (r)) = 0. (4.33) For a Schwarzschild spacetime in either Schwarzschild or ingoing Eddington–Finkelstein coordinates, this results in B r (r) = 64C , r2 (4.34) where C is an integration constant which is determined by assigning a value to B r at the critical radius. Bcr = B r (rc ) = 64C . rc2 (4.35) We use the plasma beta, βp , to parametrize the magnetic field amplitude: βp = 2P . b2 (4.36) Recall from (2.99) that b2 = gµν bµ bν , where bµ = (bt , bj ) = W Bi (γij v i B j ), + αbt vˆi . α W Given that bµ = (bt , br , 0, 0), as defined by (2.99), we have, b2 = gtt (bt )2 + gtr bt br + grr (br )2 . (4.37) Using Schwarzschild coordinates, we have gtr = 0, which simplifies the expression. To determine the magnitude of the magnetic field at the critical point, we solve for C: 2Pc (βp )−1 = 1− 2M rc (bt )2 + 1 − 2M rc −1 (br )2 = 1− 2M rc 2 (Bcr ) . (4.38) This configuration cannot be reached using an arbitrary initial condition due to the one dimen129 4.6. STEADY STATE ACCRETION sional nature of this steady state problem. The magnetic flux is exactly zero in the closed form solution; therefore, we cannot have a time dependent magnetic field. Despite the fact that this problem is not physical, it does prove to be a useful test to determine the limitations of a numerical scheme, as such this set-up is only being used for code evaluation. Examination of the closed form of the system of equations shows that the magnetic field of this form does not change the equations of motion. Figure 4.17: Magnetized relativistic spherical Bondi accretion. The results shown are at time t ∼ 100M . The left column is the result of plasma βP = 10, the right column is βP = 1. The top shows the radial velocity, the bottom the thermodynamic pressure. We see that we experience difficulty when trying to evolve fluids with a smaller βP , although the profile of the flow variables remains similar, the magnitudes all disagree with the exact solution. With higher resolution this problem begins to decrease; however, since we are ultimately interested in two dimensional simulations the resolutions demands are too great for the problem to be tractable for small βP . Larger βP returned results similar to the βP = 10 case. 130 4.6. STEADY STATE ACCRETION Figure 4.18: Convergence of dv r /dt for the relativistic magnetized spherical accretion problem. density. The convergence of the time derivative of the radial velocity. We see that the solution is converging to zero. As with the hydrodynamic spherical accretion problem we see that the solution is converging to the steady state, dv r /dt = 0. Only 5 ≤ t ≤ 16 is shown for emphasis. 131 4.6. STEADY STATE ACCRETION Although the magnetic field flux is identically zero in closed form, we will require the use of a constraint enforcing technique due to the numerical errors, especially when larger magnetic fields are studied. When larger βp was considered, it was found that the magnetic field constraint could be freely evolved; however, for smaller βp , the magnetic field constraint violations were much more obvious and without enforcement the system behaved unpredictably. Tests show that there are limitations on the magnitude of the magnetic field that can be evolved using our code. Therefore, in this study we will need to restrict our attention to plasma beta, βP > 1. 132 CHAPTER 5 RESULTS We investigate the relativistic Bondi–Hoyle problem in the ultrarelativistic and magnetohydrodynamical fluid background. We present new results for 6 different configurations. 1. Axisymmetric ultrarelativistic hydrodynamic accretion onto an a = 0 black hole. • The fluid flow at infinity is uniform in the zˆ-direction. • The internal energy of the fluid is sufficiently large that we may neglect the rest mass density. • The fluid is flowing past a stationary, non-rotating black hole. 2. Axisymmetric ultrarelativistic hydrodynamic accretion onto an a = 0 black hole. • The fluid is as described above, but the black hole is rotating with a rotation axis aligned with the asymptotic fluid velocity. 3. Infinitely thin-disk ultrarelativistic hydrodynamic accretion onto an a = 0 black hole. • The fluid is similar to the first case except that it is restricted to flow in the equatorial plane. • The fluid is flowing past a stationary, non-rotating black hole. 4. Infinitely thin-disk ultrarelativistic hydrodynamic accretion onto an a = black hole. • The fluid is similar to the previous case except that it is flowing past a rotating black hole with an axis of rotation orthogonal to the fluid plane. 5. Axisymmetric magnetohydrodynamic accretion onto an a = 0 black hole. • The fluid is now embedded within a magnetic field that is uniform and parallel to the zˆ-direction at infinity. • The rest mass density contributes to this fluid. 133 CHAPTER 5. RESULTS • The fluid is flowing past a stationary, non-rotating black hole. 6. Axisymmetric magnetohydrodynamic accretion onto an a = 0 black hole. • The fluid is the same as above, but this time flowing past a rotating black hole whose rotation axis is parallel to the zˆ-direction. The ultrarelativistic studies show a range of fluid flow patterns, referred to as the flow morphology, that differs from the previously studied hydrodynamic flows for some parameters. In particular, when Font et al.[1], [2], and [3], hereafter referred to as FI981, FI982, and FI99 respectively, studied a similar setup, they considered the relativistic set of hydrodynamic equations of motion including the contributions of the baryon rest mass density. In our study we assume that the rest mass density is negligible when compared to the internal energy of the fluid and we neglect any conservation of the baryon density. These radiation fluids are expected to exist in the early universe between 10−6 s and 1011 s after the big bang [157], as well as inside relativistic jets25 . Our studies present a new condition, where we include the magnetic field contributions in the axisymmetric configuration. Including the evolution of the embedded magnetic field makes our results more general, and perhaps more broadly applicable to astrophysical phenomenon than existing models. Currently, the Bondi–Hoyle accretion model has be used to model accretion in three main scenarios, accretion during binary inspiral, accretion onto compact objects travelling through the interstellar medium, and accretion onto the active galactic nuclei (AGN) [38, 8]. All models to date have focused on hydrodynamic accretion, where the presence of magnetic fields are entirely neglected. This assumption is concerning, when considering that most of the matter in the universe is in a plasma state and the presence of even a weak magnetic field can be amplified through dynamos. As stated in the introduction there are several physical applications for the relativistic Bondi– Hoyle accretion setup. If we focus on the binary black hole merger studied by Farris et al. [72] we see from their picture that the background fluid is assumed to exist through the entire simulation. The origin of the hydrodynamics background is still ambiguous, and could easily be assumed to be a magnetofluid, since from a cosmological view, the background fluid contains a (weak) magnetic field [158]. If we assume that the background fluid is produced by a merger of two neutron stars, were we expect the formation of an accretion disk around the resulting compact object, where 25 The author would like to thank Charles Evans for pointing out this application. 134 CHAPTER 5. RESULTS for our case we restrict attention to the situation where the resulting compact object is a black hole. We see that the resulting black hole will travel through a magnetized background fluid. Furthermore, if we consider the binary inspiral of a black hole and a neutron star we expect the presence of a magnetic field originating from the neutron star. Giacomazzo et al. determined that realistic magnetic fields in merging neutron stars will not be sufficiently large to be detectable during the merger [159]. However, they are important after merger due to effects like the Kelvin–Helmholtz instability which amplify the field strength [159]. The effects of the magnetic fields after the merger do have an impact on the accretion luminosity and relativistic jets that result from the merger [160]. As presented in Farris et al. [72], by studying relativistic Bondi and Bondi–Hoyle accretion onto binary mergers were are able extract both a gravitational wave signal as well as an electromagnetic signal from the merger. Farris performed this study using an ideal hydrodynamic fluid background, and assumed the presence of a background magnetic field was dynamically unimportant. A more general approach will need to consider the effects of the accretion of a magnetohydrodynamic fluid background onto the binary. This is particularly important for considering the effect of the large local magnetic fields that are expected to be generated after the merger [159]. Magnetic fields are known to be present in the interstellar medium of spiral, bar and irregular galaxies [161]. Consequently, we cannot neglect the presence of the magnetic field threading the background fluid accreting onto an object travelling through interstellar material if we want to make accurate models. As we show below, the presence of even a weak magnetic field can have a measurable impact on the dynamics of a system. The ultrarelativistic study is useful for understanding the early universe where it is expected that primordial black holes will form [157, 162]. There has been debate over the potential modern day observations of primordial black holes. There have been arguments to suggest that such black holes would have accreted a sufficient amount of matter to overcome Hawking radiation and be no different from solar-mass black holes in the modern era [163]. However, there has been other arguments put forward that when one includes additional physics, originally ignored, that the accretion rates are insufficient to overwhelm the Hawking radiation, and that the primordial black holes would be observable today [164, 165]. Our study indicates that the accretion rates of the primordial black holes is insufficient to evolve them to solar-mass black holes. 135 5.1. ACCRETION PHENOMENON AND ACCRETION RATES As a technical note we point out that each of the simulations took a minimum of 4 days using 64 processors. This is equivalent to 256 days using a single processor, in essence without parallelization this study would not be possible or at least confined to a small range of parameters. When we study larger spatial domains the simulations needed to run longer and could take over a week using the same resources. Before we proceed to present our results, we introduce the mathematical description of the measurements made. We consider monitoring different accretion rates including mass accretion, energy accretion, and momentum accretion rates, described below. 5.1 Accretion Phenomenon and Accretion Rates To determine the behaviour of the accretion solution, we consider the accretion rates for several different fluid quantities into the black hole. In particular, we consider the accretion rates for the energy, the radial momentum, the angular momentum, and, when studying the GRMHD system, the mass [8, 1, 2, 3, 95]. In the cases where the magnetic field is embedded in the background fluid, we also consider the energy and momentum accretion of the magnetic fields alone. Below we describe the methods of calculating these properties for any given system. 5.1.1 Rest Mass Accretion Rate The rest mass accretion rate is calculated from the gauge invariant mass [1, 2]. This quantity is derived from the volume integral of the relativistic density of the fluid, D = D(t, r, θ) over a given proper volume, V: √ γDdV . m(t) = (5.1) V If we differentiate with respect to time and then substitute the conservation of baryon number (2.87), we have an expression for the baryon accretion rate, across a closed surface: dm =m ˙ = dt V √ ∂t γDdV = − V √ ∂i −gDv i dV = − √ −gDv i dSi . (5.2) We would then scale this expression with the mass per baryon. In our case we choose this mass to be unity, leaving the expression unaltered. In the event that the fluid accretes in a steady state the closed integral may be evaluated over any spatial surface in the domain. For simplicity, we 136 5.1. ACCRETION PHENOMENON AND ACCRETION RATES use the surface defined by the Schwarzschild radius, Rs = 2M, for all measurements, since this point is the same for all equally massive black holes studied in this survey. This will allow us to compare the accretion rates for all black holes in our survey. We use geometric units in which we set G = c = M = 1, where M is the mass of the black hole. 5.1.2 Stress-Energy Accretion Rates In relativistic systems, Killing vectors arise from symmetries and imply the existence of conserved quantities, as discussed in Chap. 2. The conserved quantities we are interested in are determined by, Q(ι) (t) = V where √ −gT µν (ι)ξµ nν dV , (5.3) (ι) ξµ are the Killing vectors, nν is the 3+1 normal covector as seen in (2.12), and V is some proper volume of spacetime. In the case of a Kerr and Schwarzschild spacetimes, we have a temporal and an azimuthal Killing vector. The temporal Killing vector indicates that the energy of the system is conserved and the azimuthal indicates a conservation of angular momentum. 5.1.3 Energy Accretion Rate Using the temporal Killing vector, E(t) = V (t) ξµ in (5.3) we obtain √ −gT tt αdrdθdφ = √ E −g 2 αdrdθdφ = α V √ γEdrdθdφ, (5.4) V by differentiating with respect to time we get the energy accretion rate through some spatial surface. We are specifically interested in the energy accretion rate onto the black hole, so we define our surface to be the surface defined by the Schwarzschild radius: dE (t) = E˙ = dt √ ∂t ( γE) drdθdφ V = ∂i V = ∂V √ −g Eˆ v i + P v i − αbt B i /W drdθdφ + √ −g Eˆ v i + P v i − αbt B i /W dθdφ + V √ V √ −g T µt α,µ − T µν Γtνµ drdθdφ −g T µt α,µ − T µν Γtνµ drdθdφ. (5.5) 137 5.1. ACCRETION PHENOMENON AND ACCRETION RATES Here we use Eqn. (2.96) to express the temporal energy derivative in terms of the energy flux and the geometric source term, and recall that we define vˆi = v i − β i /α. We also use Gauss’ theorem to convert the volume integral of the divergence of the flux to a surface integral. ∂V denotes the surface of the volume V that is defined by the Schwarzschild radius. When we discuss the energy flux in the ultrarelativistic system we set B i = 0. 5.1.4 Azimuthal Angular Momentum Accretion Rate The angular momentum transfer from the fluid background onto the black hole is defined in a similar fashion as the energy accretion rate. In this calculation we use the azimuthal Killing vector (φ) ξµ , which gives us dQ(φ) = P˙φ (t) dt = − ∂V √ −gT r φ dθdφ + V √ −gT µν gνφ,µ − Γλ µν gλφ drdθdφ. (5.6) We use the azimuthal component of Eqn. (2.95) to describe the temporal derivative of the azimuthal angular momentum in terms of the divergence of the flux, then use Gauss’ theorem to convert the volume integral of the divergence of the flux to a surface integral. In each of the accretion rates defined above we have a flux contribution that is an integral over the surface of a volume which we will denote T1 , and a volume integral of the geometric source terms, denoted T2 . All accretion calculations then have the form P˙j = T1 + T2 . (5.7) The spatial volume over which we take this integral is the volume between the event horizon, rEH and the radial location of which we make our calculation, Rs . The second term is π 2π Rs (Source) dV . T2 = 0 0 (5.8) rEH We have used “Source” to represent the expression inside the volume integral from Eqns (5.5) and (5.6). 138 5.1. ACCRETION PHENOMENON AND ACCRETION RATES 5.1.5 Radial Momentum Accretion Rate As the black hole travels through the fluid background it focuses the matter not immediately accreted behind it by the gravitational interaction between the matter and the black hole. The accumulated matter could begin to have its own gravitational effect on the black hole. This would be realized through gravitational drag. Chandrasekhar referred to this effect as dynamical friction [166, 167, 168, 169, 33]. In previous papers by Font et al. [2, 3] relativistic drag has been defined to be P˙r (t) = − =− ∂t V √ ∂V √ −gT t r drdφdθ −gT r r dθdφ + V (5.9) T µν gνr,µ − Γλ µν gλr dV , (5.10) where V is the proper volume enclosed by the proper surface ∂V. Unfortunately, this is a misconception, as this calculation would necessarily require that in the asymptotic limit we have a radial Killing vector, and even then this would not correspond to a relativistic drag. While referring to this calculation as a measure of drag is unphysical, this integral does provide another calculation that may expose instabilities in the flow. We perform this calculation so that we may compare our results to those from previously published works. We refer to the results from Eqn. (5.10) as the radial momentum accretion rate, but caution the reader that no physical information can be extracted from this calculation. Rescaling The various accretion rates in this chapter were rescaled with respect to the asymptotic Bondi accretion rate. Petrich et al. [8] calculated their accretion rates at the accretion radius, and scaled their calculated accretion rates by using the Bondi accretion rate as evaluated at the transonic radius. Since our measurements were recorded at the Schwarzschild radius we needed to use a different scaling value. For each geometric configuration described below, the calculation was repeated using v∞ = 0 which corresponds to the static Bondi accretion problem. For the cases with rotating black holes the rescaling factor was taken to be the Bondi solution for the non-rotating black hole, as the Bondi accretion problem does not take into account the rotation of a black hole. This ends our description of the accretion rates studied. We now present the initial conditions and results for the different geometric configurations introduced in Chap. 2. First, we investigate 139 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE the ultrarelativistic hydrodynamic (UHD) systems in four different configurations: • axisymmetric accretion onto a spherically symmetric black hole, (a = 0, Γ = 4/3, φ = const); • axisymmetric accretion onto a axisymmetric black hole, (a = 0, Γ = 4/3, φ = const); • non-axisymmetric infinitely thin-disk accretion onto a spherically symmetric black hole, (a = 0, Γ = 4/3, θ = π/2); • infinitely thin-disk accretion onto an axisymmetric black hole, (a = 0, Γ = 4/3, θ = π/2). Recall that we define Γ to be the adiabatic constant, and that the black hole is parameterized by its mass M , which for our purposes is set to unity, as well as its spin parameter −1 < a < 1. When we state φ = const it means that we are describing the space time using only the radial and polar coordinates (r, θ) and choosing to set the φ coordinate to be constant. In axisymmetric systems setting φ = 0 is a matter of convenience, and is not a restriction on the system. When θ = π/2 we are describing the matter in spacetime using the radial and azimuthal coordinates (r, φ), and holding the polar coordinate fixed at π/2, so all matter is in the equatorial plane of the black hole. Unlike axisymmetry, holding θ fixed is a restrictive assumption. In the last part of this chapter, we discuss the axisymmetric accretion of a magnetohydrodynamic fluid: • axisymmetric accretion onto a spherically symmetric black hole, (a = 0, Γ = 4/3, 5/3, φ = const); • axisymmetric accretion onto a axisymmetric black hole, (a = 0, Γ = 4/3, φ = const). In all simulations we will choose a set of asymptotic velocities for the fluid flowing past the black hole. The values used are referenced in the appropriate sections. 5.2 Axisymmetric Bondi–Hoyle UHD Accretion Onto a Black Hole In this configuration, our state variables q as defined in Eqn. (2.122) take on the coordinate dependence, q = q(t, r, θ). See Fig. 5.1 for a graphical representation of this geometry. 140 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE zˆ r θ Figure 5.1: The axisymmetric Bondi–Hoyle accretion setup. The radial coordinate increases away from the “centre” of the black hole. The angular coordinate increases from top (θ = 0) to bottom (θ = π). The arrows inside of the half-disk show the direction of fluid flow. This geometry is valid for any φ = const. The portion of the diagram in the black shaded region is inside the event horizon/black hole. The flow is in the positive zˆ direction. We use inflow boundary conditions at rmax for π/2 ≤ θ ≤ π. For the boundary rmax , 0 ≤ θ < π/2 we use extrapolated outflow boundary conditions. The boundary at rmin is also treated using extrapolated outflow. The angular boundaries θ = 0 and θ = π use reflective boundary conditions. 141 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Initially, an infinite uniform fluid medium surrounds the black hole. We parametrize the properties of the surrounding fluid by the asymptotic speed, v∞ , the adiabatic constant Γ, and the i asymptotic speed of sound c∞ s [8, 1]. We write the initial conditions for the fluid 3-velocity, v in the Kerr–Schild coordinates as [1]: v r (0, r, θ) = 1 √ v∞ cos θ; grr (5.11) v θ (0, r, θ) = 1 − √ v∞ sin θ; gθθ (5.12) v φ (0, r, θ) = 0. (5.13) Here gθθ = gθθ (r) and grr = grr (r) are elements of the metric tensor. This velocity configuration corresponds to a black hole travelling with a velocity v∞ in the Cartesian zˆ-direction, as seen by an observer at infinity. We can easily verify that the square of the velocity as expressed above 2 reduces to v 2 = v∞ . We define our speed of sound to correspond to the ultrarelativistic system (2.58) [36, 10, 1]: c∞ s = √ Γ − 1. (5.14) Although the speed of sound is constant through the entire UHD domain, we keep the superscript denoting the asymptotic value for consistency with other works such as those presented in FI981. Since the characteristics of the system are independent of the pressure, the solution to the equations of motion will scale with the pressure. Thus, without loss of generality, we take P∞ = 1. When the simulation is turned on, the system undergoes a transition period until a steady state is reached. Once this transition time has passed, we expect to see the various accretion rates become constant if the flow stabilizes. For the radial boundary condition in the upstream side of the domain (r = rmax , π/2 < θ ≤ π), we used an inflow boundary condition. Recall that the inflow condition holds the values of all variables constant to the values prescribed at initialization. For the radial boundary located inside the event horizon, r = rmin as well as in the downstream region (r = rmax , 0 ≤ θ ≤ π/2) we used extrapolated outflow for all state variables. To treat the polar boundaries numerically, we used reflection symmetry, also known as a wall boundary condition. The techniques for evaluating these boundaries are described in Chap. 3. We also note that the value of rmax is twice as large as those considered in Font et al. [1]; this was necessary to capture the physics while ensuring the solutions 142 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Model U1 U2 U3 U4 v∞ 0.3 0.6 0.7 0.9 M 0.5 1.0 1.2 1.6 MR 0.4 1.1 1.4 2.9 Table 5.1: The parameters used for systems studied in the ultrarelativistic (a = 0, Γ = 4/3, φ = const) regime. We consider four specific cases in detail, when the flow is asymptotically subsonic (U1), the flow is marginally supersonic (U2), and when the flow is supersonic (U3 and 1/3 ∼ 0.57. The third column U4). The adiabatic constant Γ = 4/3, corresponds to c∞ s = records the asymptotic Mach number for the flow, and the last column records the relativistic Mach number as described in Eqn. (2.128). of the equations of motion are not being adversely affected by boundary conditions. For the flow speed well above the speed of sound for all models investigated, the size of the domain did not adversely affect the simulation. The change of domain size was essential for capturing the bow shocks present in the marginally supersonic flows. Further tests will be necessary to determine if the bow shocks are present in the marginally supersonic flows for the original model studied by Font. All simulations were performed using 400 × 160 resolution for 105 time steps, utilizing parallel techniques via PAMR [170] and the message passing interface (MPI)26 to increase the speed of the individual runs. We use the domain rmin ≤ r ≤ rmax , 0 ≤ θ ≤ π. All ultrarelativistic fluids are parameterized using parameters the spin parameter of the black hole, a, the asymptotic speed of the fluid, v∞ , and the adiabatic constant, Γ. In our study we restrict our attention to Γ = 4/3 fluids as mentioned in Sec. 2.7. 5.2.1 Axisymmetric Accretion: a=0 We perform a parameter survey over the asymptotic flow velocity, v∞ , in the ultrarelativistic regime, investigating the parameters listed in Table 5.2. The previous work presented in FI981 considered the mass accretion rate as one of the measures to determine a steady state flow. We consider a system in which the baryon rest mass density, ρ0 , is negligible compared to the energy of the fluid, so we are not able to directly measure the mass accretion rate. Instead, we take advantage of the timelike Killing vector and measure the energy accretion rate. When we simulate model U2 in Table 5.1, we end up with a system where the tail shock, which is present for a brief period during the transient solution, detaches from the black hole as 26 https://computing.llnl.gov/tutorials/mpi/ 143 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE the high pressure region in the downstream region slowly migrates around to the front and makes contact with the axis of symmetry. By performing convergence testing and using different boundary conditions, we determined that this was a physical feature of the simulation. We refer the reader to the appendix Fig. A.3 for a time evolution of this system. Model U2 develops a steady bow shock just beyond the upstream side of the black hole. Although some of the fluid falls into the black hole, there is a pressure build-up around the black hole, most dominantly on the downstream side. On the upstream side this resembles the spherical accretion as seen in Chap. 4. As the simulation proceeds the fluid pressure downstream overwhelms the fluid flowing on the upstream side and begins to wrap around the black hole, forming a pressure build-up in the upstream side as seen in the appendix in Fig. 5.8. In the moderately supersonic and subsonic cases, the built-up pressure eventually overcomes the pressure from the fluid at the front of the black hole, pushing the fluid back upstream. This is displayed in figure Fig. A.3 for model U2. If the black hole travels at subsonic speeds, such as in model U1, an initial wave propagates out of the system, leaving the domain we calculate, and the resulting flow reaches a steady state, with no shock. When the black hole has a faster asymptotic velocity, as in models U3 and U4, we again have a tail shock and the system behaves much like the hydrodynamic counterparts investigated by FI981 [1]. See Fig. A.4 for a time evolution of model U4. For model U4, the fluid flowing past the black hole puts enough pressure on the matter behind the black hole to maintain an accretion column. A cross section of the pressure profile may be seen in figure 5.2, where we see that the pressure profile upstream does not appear to smoothly reach the upstream boundary condition. To ensure that this propagation of internal effects to the boundary does not alter the interior physics, we repeated the simulation using a value of rmax = 100. Using the larger maximum radius, the upstream and downstream profiles show that the pressure at the boundary matches the asymptotic pressure. The resulting flow is stable. The steady state pressure profile may be seen in Fig. 5.9. The morphology of our results for models U3 and U4 agree with the behaviour of the relativistic hydrodynamic accretion process observed in FI981 [1]. The resulting energy and radial momentum accretion rates for models U2, U3, and U4 are seen in Fig. 5.3. A convergence test is performed using slices through the domain across defining features such as shock fronts or the angle of attachment of the tail shock to the black hole. The results of these tests for model U4 and U2 may be seen in Figs. 5.5 and 5.6 respectively. These tests 144 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE indicate that the code used to simulate the fluid evolution converges, with respect to resolution, to a physical solution. Fig. 5.5 shows that as the resolution of the simulation is increased, the opening angle of the Mach cone converges to a fixed value, a requirement of a steady state accretion flow. Fig. 5.6 converges, with resolution, to a bow shock. The convergence tests presented indicate that the features in the fluid are not merely numerical artifacts, but real physical features of the system under investigation. A snapshot of the evolution of the pressure field at time t = 1000 for velocities v∞ = 0.6, 0.9 may be seen in Figs. 5.8 and 5.9. We see in Fig. 5.8 the bow shock present in the upstream region of the calculated domain. This shock front stabilizes and reaches a steady state. In previous simulations such a phenomenon was not present [1]. It was speculated that the formation of a bow shock was a necessary condition for the presence of instabilities, which did not occur in previous studies. We have found a system (model U2) where bow shocks are present. Although model U2 was determined to reach a steady state, a wider parameter survey may reveal other similar systems that do experience instabilities. For v∞ = 0.3, the flow produced a wave travelling upstream, which interfered with the upstream boundary conditions, this simulation was terminated earlier at t = 300M . Models U3 and U4 exhibit similar flow morphologies as seen in [1]. As we have shown for some parameters in the ultrarelativistic model a bow shock develops. The bow shock is a feature not seen in previous hydrodynamic models studied by Font et al.There are two possible reasons for our results differing from those presented by Font et al.; first, we study a different physical model with very large energy densities. It is possible that the resulting bow shock is due to the extreme energy of our system. The second possibility to explain the differences in the results is that we are studying a different set of parameters than those studied in Font’s papers. Font et al. studied flow with speeds well above the speed of sound, while we include parameters that are only slightly above the speed of sound. Further studies are necessary to determined if the bow shock is possible in the original model studied by Font et al., but we do not pursue this comparison in this thesis. We have also shown that the study of moderately supersonic systems required the need for a larger domain of integration as compared to previously investigated models. We cannot say if extending the domain of integration using the original model will also allow us to capture a bow shock. We leave answering this question to a future investigation. 145 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Figure 5.2: Ultrarelativistic pressure profile in the upstream, θ = π/2, (left column) and downstream, θ = 0, (right column) regions for model U3 (top) and U4 (bottom). We see that the pressure in the upstream region of the black hole is small by comparison to the pressure in the wake in the downstream region. The image on the left is magnified to show that there is a slight increase in pressure as the fluid reaches the black hole. As the asymptotic velocity of the black hole increases the upstream pressure deviates less from the pressure defined at the boundaries. Since the upstream pressure profile does not smoothly agree with the upstream boundary condition, a larger domain was required for simulations of slower black holes. 146 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Figure 5.3: Axisymmetric ultrarelativistic energy accretion (top) and radial momentum accretion (bottom) rates for Γ = 4/3 for models U2, U3, and U4. The faster the black hole is travelling the more energy it accretes, and the larger the radial momentum accretion rate becomes. This is in agreement with the results from Font et al. [2]. 147 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Figure 5.4: Ultrarelativistic pressure profile upstream (top) and downstream (bottom) for model U2 with radial domain rS ≤ r ≤ 100. We see that the pressure in the upstream region of the black hole is small by comparison to the pressure in the downstream region, or wake. The pressure profile in the upstream region indicates that there is a shock in the upstream region of the black hole. This detached shock is a bow shock that forms when the black hole is travelling at marginally supersonic speeds, v∞ ∼ c∞ s . This bow shock does not exist when we investigate black holes with a larger asymptotic speed as seen in the upstream profile for the black hole with v∞ = 0.9, as seen in Fig. 5.9. The above profiles were taken at t = 1000M . With the larger radial domain, we see that the pressure profile smoothly matches the upstream boundary conditions. The dashed line in the downstream profile emphasizes the maximum pressure of the fluid surrounding the black hole on the upstream side. 148 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Figure 5.5: Ultrarelativistic pressure profile at the event horizon for the axisymmetric evolution model U4 for three different resolution levels. We see that the attachment angle at the event horizon is converging on the solution θ ∼ 1.4. The resolution changes are 4 : 2 : 1. The lowest resolution (level 1) is Nr × Nθ = 200 × 80. For our convergence testing, we simultaneously scale both spatial dimensions by two for each level, whereas previous authors only consider scaling the resolution of the angular dimension. 149 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Figure 5.6: Ultrarelativistic pressure profile in the upstream region for the axisymmetric evolution model U2 for three different resolution levels. We see that the front of the bow shock converges to r ∼ 13 at time t = 1000M . The resolution changes are 4 : 2 : 1. The system is converging on a solution. The lowest resolution (level 1) is 200 × 80. Only the radial domain 2 ≤ r ≤ 20 is plotted to show more detail. 150 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Figure 5.7: Ultrarelativistic pressure profile for model U1. This configuration does not have a shock wave, which is in agreement with other subsonic studies [8, 1]. Model U6 U7 U8 U9 U10 U11 a 0.5 0.9 0.99 v∞ 0.6 0.9 0.6 0.9 0.6 0.9 M 1.04 1.56 1.04 1.56 1.04 1.56 MR 1.06 2.91 1.06 2.91 1.06 2.91 Table 5.2: The parameters used for systems studied in the ultrarelativistic (a = 0, Γ = 4/3, φ = const) regime. We consider two specific cases in detail, when the flow is marginally supersonic (v∞ = 0.6), and when the flow is supersonic (v∞ = 0.9). The adiabatic constant Γ = 4/3, 1/3 ∼ 0.57. As in table 5.1 we record the Mach numbers corresponds to the speed of sound c∞ s = for reference. 5.2.2 Axisymmetric Accretion: a=0 In this series of simulations, we extend the axisymmetric accretion problem to allow the black hole to rotate along the axis of the flow. Accretion onto an axisymmetric rotating black hole has been performed in the specialized case for the adiabatic constant Γ = 2 by Petrich [38]. For Γ = 2 fluids the speed of sound is unity, and thus supersonic flows for these fluids are impossible. Other studies are mentioned in several papers [8, 1], but we have found no evidence that this problem has been studied numerically for less stiff fluids i.e. smaller Γ. Polytropic fluids with an adiabatic constant Γ = 2 are commonly used for some neutron star models [10]. In this thesis, we are considering fluids with large internal energy, but not sufficiently high that c∞ s = 1. 151 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Figure 5.8: Ultrarelativistic pressure profile onto a spherically symmetric black hole. This image is a marginally supersonic evolution model U2. Note the presence of the bow shock close to the front of the black hole. The profile of the bow shock is investigated in more detail in Figs. 5.4 and 5.6. The initial conditions as presented in (5.2) can also be used for a fluid around a rotating black hole. We investigate the parameters used in the previous section for models U2 and U4, with the addition of a rotation parameter taking the values of a = 0.5, a = 0.9, and a = 0.99. These are summarized in Table 5.2. For the parameters investigated, we found that the accretion rates reach a steady state. The energy accretion rates for models U6, U8, and U10 may be seen in Fig. 5.13, while the energy accretion rates for models U7, U9, and U11 are seen in Fig. 5.14. We see that the higher velocity configurations reach a steady state solution faster, which is in agreement with the studies performed with a = 0, as well as the hydrodynamic non-rotating study performed in FI981. Just as in FI981 we find that the accretion rates oscillates around some central value, but as the resolution increases the amplitude of the oscillations diminish. We turn the reader’s attention to Fig. A.5 for a time evolution of model U7, and Fig. A.7 for model U11. In Fig. 5.10 we see a cross-section of the pressure field at θ = π (upstream), and θ = 0 (downstream). We compare the upstream and downstream profiles for models U4, U7, and U9. The effects of rotation are only truly distinguishable from models with a = 0 for models with spin parameter a > 0.5. When we consider a cross-section through r = 4 in Fig. 5.11, we compare 152 5.2. AXISYMMETRIC BONDI–HOYLE UHD ACCRETION ONTO A BLACK HOLE Figure 5.9: Ultrarelativistic energy accretion onto a spherically symmetric black hole. Here we have a snapshot of the black hole in model U4 at time t = 1000M . The black hole is in the highly supersonic regime. This plot shows evidence of the typical tail shock experienced in this type of flow [1]. Figure 5.10: Ultrarelativistic pressure profiles for the rotating axisymmetric accretion problem for v∞ = 0.9, models U4, U7, U9. We see that for black hole spin parameters a = 0 and a = 0.5 the downstream (bottom) pressure profiles are only marginally distinguishable. As the spin is increased to a = 0.9, we see a much larger pressure on the upstream region of the black hole (top). The profiles on the downstream region become slightly sharper when the spin parameter in increased. 153 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.11: Ultrarelativistic axisymmetric pressure profile P (1000, 4, θ) for a section through the tail shock at r = 4M and asymptotic velocity v∞ = 0.9, models U4, U7, U9. We see the opening angle of the tail shock widens as the spin parameter in increased. Just as in the pressure profile along the θ = 0, π lines are altered by the spin parameter. The opening angle changes slightly for a spin a = 0.5 (dashed line) black hole as compared to a spin a = 0 (dotted line) black hole, but when we consider an a = 0.9 (solid line) black hole we see a larger shock opening angle. We also see that as the spin increases the pressure profile on r = 4M becomes smoother and the amplitude decreases. the opening angles of the tail shock. As with the upstream and downstream profiles, we can only marginally distinguish the cross-section for models U4 and U7, while model U9 has a wider opening angle. Convergence tests for this geometry may be seen in figure 5.12. This concludes our work in axisymmetric ultrarelativistic accretion onto a black hole. We now turn our attention to the non-axisymmetric infinitely thin-disk model introduced in Sec. 2.8. 154 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.12: Ultrarelativistic axisymmetric pressure profile P (1000, 4, θ) for a section through the tail shock at r = 4 for model U9. We see the opening angle of the tail shock converges to θ ∼ 1.2. The pressure tends to a maximum of Pmax ∼ 36. Model U12 U13 v∞ 0.6 0.9 M 1.04 1.56 MR 1.06 2.91 Table 5.3: The parameters used for systems studied in the ultrarelativistic (a = 0, Γ = 4/3, θ = π/2) regime. We consider two specific cases in detail, when the flow is marginally supersonic (U12), and when the flow is supersonic (U13). The adiabatic constant Γ = 4/3, corresponds to 1/3 ∼ 0.57. The last two columns record the Mach number and relativistic Mach number c∞ s = for the flow. 155 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.13: Energy accretion rates, models U2, U6, U8, U10, for the axisymmetric configuration. We see that the energy accretion rate as measured at the Schwarzschild radius actually increases for an a = 0.5 black hole and decreases for larger spin parameters. 156 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.14: Energy accretion rates, models U4, U7, U9, U11, for the axisymmetric configuration. These results are in agreement with those seen in Fig. 5.13. We see that the energy accretion rate of an a = 0.5 black hole is a maximum out of the parameters considered in this study. This is directly related to the fact that the spin a = 0.5 black hole has the widest attachment angle and the largest wake when compared to other spins considered. 157 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE yˆ r φ x ˆ Figure 5.15: The non-axisymmetric infinitely thin-disk accretion setup. The radial coordinate increases away from the “centre” of the black hole. The angular coordinate increases from top to bottom. The arrows inside of the disk show the direction of fluid flow as viewed by an observer at infinity. The boundary of the shaded region in the centre denotes the event horizon, the area inside the event horizon is a black hole. The boundary at rmax and π/2 ≤ φ ≤ 3π/2 are treated with inflow boundary conditions. The remaining radial boundaries are treated using extrapolated outflow. The angular boundaries at φ = 0 and φ = 2π are treated numerically with periodic boundary conditions as described in Chap. 3. When referencing the geometry in the future we refer to the region below the x ˆ-axis (π/2 ≤ φ ≤ 3π/2) as upstream, and the region above the x ˆ-axis (0 ≤ φ ≤ π/2 and 3π/2 ≤ φ ≤ 2π) as downstream. 158 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE 5.3 Non-axisymmetric Infinitely Thin-Disk Ultrarelativistic Accretion Onto a Black Hole In this configuration the black hole is travelling through an infinitely thin uniform fluid background. The orientation of the black hole is such that the rotation axis is orthogonal to the direction of travel, as seen in Fig. 5.3. The accretion flow is assumed to only occur on the equatorial plane, θ = π/2. This is not expected to be a physically general setup. One would typically expect the fluid to bulge near the black hole due to extreme pressure build-up in that region. This effect is taken into consideration in several papers written about thin-disk accretion such as those cited in the introduction. This set-up is a means to understand the full three dimensional evolution that is left for a future study. As described by Font et al. [2, 3], this is a simplifying assumption. As cited in FI99, similar Newtonian works on this problem using cylindrical coordinates may be seen in Matsuda et al. [90] and Benensohn et al. [128]. We explicitly assume that the flow has no dependence on the polar coordinate. If a flow is reflection symmetric on the equatorial plane there will be no polar pressure gradient, and the polar velocity will vanish, therein the infinitely thin disk model is justifiable. For more general configurations a full three dimensional treatment is required [3]. We initialize the variables using a similar procedure as with the axisymmetric case; however, the fluid 3-velocity, v i , will now take the form of those used in FI99: v r (t, r, φ) = F1 v∞ cos φ + F2 v∞ sin φ; (5.15) v θ (t, r, φ) = 0; (5.16) v φ (t, r, φ) = −F3 v∞ sin φ + F4 v∞ cos φ. (5.17) 159 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE We define the Fi functions written explicitly in terms of the general metric: F1 = 1 √ ; grr (5.18) F4 = −2grφ F1 ; gφφ (5.19) F3 = F2 = F1 grr + F4 grφ 2 grr gφφ − grφ ; F3 F4 gφφ + F1 F3 grφ . F1 grr + F4 grφ (5.20) (5.21) Equations (5.18) to (5.21) are a slightly simpler form of the function Fi described in Papadopoulos et al. [95]. The variables, grr , gφφ and grφ are elements of the spacetime metric and are functions of the spacetime coordinates. As with the axisymmetric case, the characteristics are independent of the thermodynamic pressure. Therefore, we will take P∞ = 1 without loss of generality. For these simulations we have restricted the system to the equatorial plane which satisfies the infinitely thin-disk approximation as described in Chap. 2.8. 5.3.1 Infinitely Thin-Disk Accretion: a=0 The plot of the energy accretion rate may be seen in Fig. 5.16, where we see the dependence of the energy accretion rate on the velocity of the black hole. We also see that the faster the black hole is travelling, the longer it takes for the system to reach a steady state. This is in contrast to the axisymmetric simulations and to the findings of FI982. We perform a convergence test for the system at v∞ = 0.6 to verify the results. Snapshots of these simulations may be seen in Fig. 5.22 and 5.23. We refer the reader to the appendix Fig. A.11 for the time evolution of model U12 using rmax = 50, and Fig. A.14 for rmax = 1000. Model U13 is seen in Fig. A.13, where we focus on the region 2 < r < 50, and Fig. A.15 for the evolution of the full domain. The evolution for models U12 and U13 developed a tail shock. The tail shock formed very early in the evolution, with the presence of highly evacuated regions (signified by very low relative pressure) on the edge of the tail. While the fluid accumulates on the black hole, a secondary wave begins to approach the front shock. In this configuration, the fluid collects against the black hole, and wraps around to the downstream side. However, unlike the axisymmetric setup previously discussed, there is a large pressure that builds in the upstream side of the black hole that pushes 160 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.16: Nonaxisymmetric ultrarelativistic energy accretion rates for models U12 and U13. We see a dependence of the accretion rate on the velocity of the black hole. more oncoming fluid back upstream. This upstream travelling bow shock is typically evidence that the tail detaches; however, in this case we see that that a tail shock remains. We conclude that the bow shock presented here results from fluid reflecting off of the matter built up in the front of the black hole. A profile of the bow shock wave is seen in Fig. 5.17. In Fig. 5.22 we see that the reflected wave extends out to the edge of the tail shock. This shock travels faster than the black hole, and will eventually reach the outer boundary of the domain. Once the shock hits the outer domain, the boundary conditions used in the simulation would be incorrect due to the fact that they were held constant. Consequently, any further evolution would become meaningless in this simulation. We see from the profiles in Fig. 5.20 that the flow pressures in the wake are much higher than those in the upstream flow. To determine the behaviour of the accretion phenomenon, we also measure the angular momentum accretion. Previous studies by Font et al. [2] in the hydrodynamic regime show that the angular momentum accretion does not attain large amplitude for any simulation. Their results 161 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.17: Ultrarelativistic pressure profile for model U13 at time t = 1000M. On the left we see the profile of a shock moving upstream, while on the right we see the downstream profile. The dashed line emphasizes the amplitude of the maximum upstream pressure. show the angular momentum accretion rate profile oscillating with very low amplitude around zero. The angular momentum accretion rates for a = 0 are seen in Fig. 5.25. We find that the angular momentum accretion rate increases as a function of resolution as seen in Fig. 5.26. The simulations performed with different radii defining the outer boundaries had very different quantitative results, which may be seen in the comparison of the two accretion rates in Fig. 5.21. The accretion process studied here must also be a function of rmax . We consider rmax = 1000, while holding the resolution constant. The results of these simulations are seen in Figs. 5.22, and 5.23. We present a preliminary study of this phenomenon, a more detailed study of the functional dependence of the accretion process on the quantity rmax is left for a future project. We compare the flow morphology for the two different radii in Fig. 5.24, and find that the morphology remains similar; however, the magnitude of the pressure and other quantities are quite different. 162 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.18: Ultrarelativistic pressure profile for the thin disk approximation for model U11 for rmax = 1000. This cut is taken at the Schwarzschild radius. The a = 0 simulation at time t = 1000M (top) shows symmetry around the black hole while the a = 0.5 and a = −0.5 simulations at time t = 1000M (bottom) shows a deviation from the symmetry due to the rotation. The regions close to φ = 0 and φ = 2π are downstream of the black hole. The a = 0 simulation shows that there is only a small pressure difference between the upstream and downstream sides of the black hole. The rotating black hole exhibits a larger pressure in the downstream region, while the pressure in the upstream part of the domain, π/2 < φ < 3π/2, is similar to the a = 0 model. 163 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.19: Ultrarelativistic infinitely thin-disk accretion a = 0. This plot shows the energy accretion rate for model U12. This image displays the result for time t = 1000M , long after the evolution has reached a steady-state. 164 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.20: Ultrarelativistic infinitely thin-disk accretion onto a spherically symmetric black hole for model U12. This is a cross section of the domain at angle φ = 0 and time t = 1000M . The pressure profile appears very similar to that of the axisymmetric profile. The same cut was performed along the angle φ = 2π to reveal the same form. In the upstream image, we see that the outer boundary is very different from the boundary condition that P = 1. When we push the boundary out further, we are able to capture more features of the accretion system before the boundary effects begin to disrupt the system. In the bottom plots we restrict our attention to smaller domains to aid in visualization. 165 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.21: Ultrarelativistic energy accretion rates for model U13 with rmax = 50, 1000, a = 0. We see from this preliminary survey that there is a dependence of the accretion rate on the maximum radius of the domain. The evolution is much smoother for the larger radius, and when we investigate the evolution of the pressure field P as seen in Fig. 5.22 and 5.23. 166 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.22: Ultrarelativistic accretion onto a spherically symmetric black hole for model U13. This is the result for P (1000, r, φ) in the infinitely thin-disk evolution with rmax = 1000. The faster the asymptotic velocity, the thinner and more well defined the tail shock becomes. The point of contact between the tail shock and the black hole decreases as a function of the asymptotic velocity. Note the presence of a bow shock radiating from the black hole. It is this bow shock that forces us to extend our simulation domain out much further than those in the ultrarelativistic axisymmetric simulation, and further than the cases considered in Font et al. [2]. The tail, magnified in Fig. 5.23, is surrounded by low pressure regions. At the time of this snapshot the tail has not reached the outer boundary, but we do note that the tail remains well defined even out to very large distances. 167 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.23: Ultrarelativistic pressure accretion onto a spherically symmetric black hole for model U13. Here, we focus on the region 2 ≤ r ≤ 50. We see a well defined tail shock. 168 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.24: A comparison between rmax = 50 (left) and rmax = 1000 (right) pressure fields for model U13. In both of these images we focus on the region 2 ≤ r ≤ 50. For the left image the pressure range is 3.3 × 10−2 ≤ P ≤ 1.3 × 102 while on the right 1.4 × 10−2 ≤ P ≤ 7.0 × 102 . We see that the pressure morphology is very similar between these two models; however, the smaller domain artificially holds the initial bow shock close to the black hole introducing unphysical waves that propagate into the black hole. The bow shock in the left image is not visible because it resides on the edge of the domain. The presence of the artificially stable bow shock explains the different range of pressures in the region displayed here. When the domain is larger the flow still develops a tail shock, but the solution is not affected by the boundary at the time when the simulation was terminated. 169 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.25: Ultrarelativistic angular momentum accretion rates for models U12 (top) and U13 (bottom) for a = 0. The amplitude of the result for model U12 is close to the tolerance parameter used to solve the system of equations and is considered to be an artifact of the numerical simulations. The result for model U13 shows a larger oscillation near the middle of the simulation time; however, since the larger oscillation damps out after a short time, the flow is still considered steady. The original study by Font et al. [1] had a similar feature and claimed that this is within the numerical error and could be dismissed. Upon convergence testing, we find that the numerical values for model U12 actually increase as the evolution proceeds, as seen in Fig. 5.26, but the amplitudes remain small. Convergence testing for model U13 indicates that the oscillations decrease with increased resolution. 170 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Figure 5.26: Ultrarelativistic angular momentum accretion rate convergence test for model U12 with a = 0. The dotted line indicates the lowest resolution (Nr , Nφ ) = [1000, 160], the dashed line represents (Nr , Nφ ) = [2000, 320], and the solid line represents (Nr , Nφ ) = [4000, 640]. We see that for each resolution the net angular momentum accretion rate remains near zero. We do note two features, first that the magnitude of the oscillation increases with resolution and second, the highest resolution oscillates around a non-zero value; however, due to the overall amplitude of the oscillation and the fact that it oscillates around a fixed value would indicate that the flow will not experience a so-called “flip-flop” instability. The oscillations presented here are too small to be seen in a visualization of the flow. 171 5.3. NON-AXISYMMETRIC INFINITELY THIN-DISK UHD ACCRETION ONTO A BLACK HOLE Another important observation is that in the presence of smooth flow the a = 0 infinitely thin-disk accretion will not experience any net angular momentum accretion. If an instability were to develop the black hole could accrete a net angular momentum from the passing fluid. In Figs. 5.25 and 5.26 we see the results of measuring the angular momentum accretion onto the black holes. We see in Fig. 5.25 that model U12 does not accrete any net angular momentum from the surrounding fluid. The fact that the momentum accretion rate oscillates around zero is indicative of numerical noise, especially when considering that the amplitude of the oscillation is only two orders of magnitude larger than numerical tolerances of the code. When we perform convergence tests as seen in Fig. 5.25, we find that the amplitude of these oscillations actually increases as a function of resolution. In particular when looking at the (Nr , Nφ ) = [4000, 6400] calculations we find the oscillations no longer oscillate around zero. This indicates that an instability may be forming, but, the amplitude of such an instability is very small. To verify this behaviour may require higher numerical accuracy methods, and at the very least requires higher resolution simulations. The verification of this instability is left as a future project, using methods described in Chap. 6. The angular momentum accreted in model U13, seen in the bottom of Fig. 5.25 exhibits a nontrivial momentum accretion rate; however, as the simulation continues the accretion rate returns to zero. Due to the geometric configuration, the only way angular momentum may be transfered to the black hole is if a tangential instability develops. We see no evidence of such an instability developing, which is in agreement with the results reported in FI982. We now describe our last parameter survey using the infinitely thin-disk approximation with a rotating black hole. 5.3.2 Infinitely Thin-Disk Accretion: a=0 When the rotation parameter is turned on, we have a non-trivial angular momentum accretion rate as seen in figure 5.27. The accretion rates indicate that the rotating black holes investigated in this study also reach a steady state following a transition period. The effects of rotation in the ultrarelativistic system was found to be in agreement with the results originally discovered by Font et al. [3]. The effects of rotation are only noticeable within a few Schwarzschild radii of the black hole. The remaining domain has very similar features to the non-rotating black hole. To check the symmetry of the system we run two separate simulations with opposite spin. The results of this simulation may be seen in Figs. 5.28. The mirror symmetry is apparent and the resulting accretion 172 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE rates reflect this behaviour. No noticeable difference in the accretion rates exists as may be seen in the energy and radial momentum accretion rate plots in Figs. 5.30, 5.32; however, the azimuthal momentum accretion rate seen in Fig. 5.31 clearly changes sign as one may expect. We note that, with rotation turned on, the instability that appeared to be developing for models U12 and U13 with a = 0 are no longer apparent. This may be due to the fact that the angular momentum accreted is much larger than that produced by the instability, so that the effects are negligible. Convergence tests reveal no evidence for the presence of instabilities. In Figs. 5.33, 5.34 we compare model U12 and U13 with a = 0 and a = 0.5. We find that the accretion rates for different quantities is a function of the spin rate. The radial momentum rates decrease as a function of spin, while the energy accretion rates increase. We should be sure not to make too strong a claim here, since the axisymmetric studies indicate that the accretion rates are non-monotonic functions of the spin rate. The axisymmetric case shows the maximum accretion rates occur when a = 0.5. As this is a preliminary result we will need to perform a larger parameter survey to find the correct relationship. This ends our discussion on the ultrarelativistic fluid regime. To see the conclusions of this work we refer the reader to Sec. 6.1.1. We now describe the results from our axisymmetric accretion of a magnetofluid onto a black hole. 5.4 Magnetohydrodynamic Bondi–Hoyle Accretion Onto a Black Hole From the work of Wald [108] and Komissarov [171], we have the electromagnetic tensor for a rotating black hole immersed in an asymptotically uniform magnetic field. The asymptotic field is aligned with the rotational axis of the black hole, defined in terms of the temporal rotational, (t) ξµ = ∂t , and (φ) ξµ = ∂φ , Killing vectors of an axisymmetric black hole: Fµν = Bo (φ) ξ[µ,ν] + 2a (t)ξ[µ,ν] . (5.22) More explicitly, we are interested in the magnetic field configuration which we may obtain from Bi = 1 ijk ǫ Fjk , 2 (5.23) 173 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.27: Ultrarelativistic angular momentum accretion rate for models U12 and U13 with a = 0.5. The trends of our accretion rates as a function of asymptotic velocity agree with the results shown in Font et al. [3]. 174 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.28: Ultrarelativistic accretion for model U13 with a = 0.5. For this simulation, rmax = 50. This image was taken at time t = 500M before the boundary effects were found to effect the accretion process. The morphology of the pressure flow rotates around the black hole. The rotational effects of the black hole were localized to a few a few Schwarzschild radii. Beyond that, the flow morphology was found to be similar to that of accretion onto a spherically symmetric black hole. The region of contact between the tail shock and the black hole decreases as a function of the asymptotic velocity; however, none of the parameters investigated exhibited a disconnection between the tail shock and the black hole. These results agrees with the findings presented in FI99. 175 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.29: Ultrarelativistic accretion for model U13 with a = ±0.5. For this simulation, rmax = 1000M , but we focus on the region 2 ≤ r ≤ 50. This image was taken at time t = 500M and we present the logarithm of the pressure to emphasize the tail shock. We see that near the black hole the pressure profile wraps around the black hole; however, a few Schwarzschild radii away and the flow recovers a profile similar to that of the non-rotating black hole. We see that the flow morphology around an a = 0.5 black hole is a mirror image to the morphology around an a = −0.5 black hole. The similarity between rotating and counter-rotating black holes is emphasized in the accretion rate plots in Figs. 5.30 and 5.32 176 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.30: Ultrarelativistic energy accretion rates for model U13 with a = ±0.5. Due to the symmetry of the flux components with respect to the rotation rate we have that the accretion rates are identical. 177 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.31: Ultrarelativistic azimuthal angular momentum accretion rates for model U13 with a = ±0.5. Due to the symmetry of the flux components with respect to the rotation rate we have that the form of the accretion rates is identical; however, the signs have been reversed. 178 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.32: Ultrarelativistic radial momentum accretion rates for model U13 with a = ±0.5. Just like the energy accretion seen in Fig. 5.30 we see that the accretion rates are identical. 179 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.33: Ultrarelativistic accretion rates for models U12 and U13 with a = 0, 0.5. We see a similar trend to the axisymmetric a = 0 simulations, the radial momentum accretion rate decreases as a function of spin up to parameter a = 0.5. Further investigation is needed to see if the trend continues to be similar to the axisymmetric case. This diagnostic calculation does not reveal any instabilities forming with the parameters investigated. 180 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.34: Ultrarelativistic azimuthal angular momentum accretion rates for models U12 and U13 with a = 0, 0.5. The energy accretion rate also increases as a function of spin up to a = 0.5. 181 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.35: Ultrarelativistic energy accretion rates for models U12 and U13 with a = 0.5. Here we again have evidence that the black hole accretes energy at a steady rate while rotating. We see that, as expected, the accretion rate depends on the asymptotic velocity of the black hole. 182 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.36: Ultrarelativistic radial momentum accretion rates for models U12 and U13 with a = 0.5. Here we have evidence that the radial momentum accreted by the black hole depends on the asymptotic velocity of the hole. The faster the black hole travels, the more radial momentum is accreted. As stated previously, we are not able to extract physical information from this quantity, but we do see that this is in agreement with the results from the Font et al.papers. 183 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.37: Ultrarelativistic radial momentum accretion rates for models U11 with a = 0 and U13 with a = 0.5. We see that for a moderately rotating black hole the radial accretion rate actually increases. This trend is independent of the asymptotic speed of the black hole, so we only show the case for v∞ = 0.9. 184 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE √ where ǫijk ≡ 1/ γ[ijk] and [ijk] is the totally antisymmetric Levi–Civita tensor. In particular, we are interested in the following magnetic field: −Bo B r = √ (γφφ,θ + 2agφt,θ ) ; 2 γ Bo B θ = √ (γφφ,r + 2agφt,r ) ; 2 γ (5.24) Bo B φ = √ (γrφ,θ + 2agrt,θ ) . 2 γ Here γij and gµν are the 3-metric, and 4-metric components, respectively, √ γ is the determinant of the 3-metric and B0 is a scaling factor that determines the magnitude of the magnetic field. This magnetic field satisfies (2.102) as it has zero divergence. When we travel far from the black hole, the rotational effects go as 1/r3 [4]27 , and the magnetic field must tend to that of a field around a spherically symmetric black hole. In that case we have gφt , grφ → 0 and (5.24) becomes, −Bo B r = √ (γφφ,θ ) , 2 γ Bo B θ = √ (γφφ,r ) , 2 γ (5.25) Bo B φ = √ (γrφ,θ ) . 2 γ To determine the more general case of an asymptotically uniform magnetic field not aligned with the rotation axis of the black hole, we refer to the work performed by Biˇca´k et al.[172], where they calculate the general magnetic field configurations as seen by an observer at infinity. Unfortunately, due to the symmetry considered in the axisymmetric problem, a tilted magnetic field (relative to the rotation axis) will break the symmetry28 . For the axisymmetric simulations, we use the Wald solution for a magnetic field parallel to the axis of rotation described in (5.24). This magnetic field configuration is sufficiently general for a spherically symmetric or axisymmetric (uncharged) black hole which is asymptotically flat. In the spherically symmetric case, the orientation of the magnetic field is parallel to the direction of black hole travel. For the axisymmetric black holes studied, the orientation of the magnetic field is parallel to the axis of rotation, which coincides with the direction of the black hole’s velocity. 27 See page 891. more detailed calculations for some geometric configurations we refer the reader to Thorne et al.[173] page 121. See also [174, 175, 172] as cited by Thorne. 28 For 185 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Model M1 M2 v∞ 0.5 0.5 Γ 4/3 5/3 O(βP∞ ) 2 1 c∞ s 0.1 0.1 Table 5.4: The parameters used for systems studied with an accreting magnetofluid with a magnetic field oriented parallel to the rotation axis. The third column is the order of magnitude selected for the asymptotic plasma beta parameter. The plasma beta parameters selected reflect the right order of magnitude for accretion disks around dwarf nova, [176], and have been used in Newtonian gravity studies of magnetized accretion disks by Hawley et al. [177, 178]. The hydrodynamic variables selected were chosen to always allow for asymptotically supersonic flow. Using Wald’s solution for models M1 and M2 we see in Figs. 5.38, 5.41, and 5.52, that the plasma beta parameter, βP (t, r, θ) = 2P (t, r, θ) , b2 (t, r, θ) (5.26) tends to values below unity near the event horizon on the downstream side of the black hole. As we noted in Chap. 2.4.3 when the plasma beta parameter, βP , drops too far below unity the MHD equations of motion are no longer valid. During the evolution of the magnetized system we found that the magnetic field on the downstream side of the black hole becomes large. This amplification corresponds to a relatively low βP (t, r, θ) in the same region. If the amplification is too great, the solution of the equations of motion become unphysical for the model considered. Numerically this would not be a problem; however, the primitive variable solver is known to have severe numerical difficulties finding a physical solution for the primitive variables when there are extreme gradients in either the velocity or the magnetic field variables. 5.4.1 Magnetized Axisymmetric Accretion: a=0 We study the MHD accretion problem onto an a = 0 black hole. The resulting evolution of model M1 may be seen at time t = 2500M in Fig. 5.38, and in Fig. 5.41 for model M2 at t = 1400M . The flow morphology for this supersonic system shows a tail shock; however, in contrast to hydrodynamic flows there is a noticeable depleted region downstream of the black hole along the axis of symmetry. In hydrodynamic flows this region has a maximum value of both the baryon density, ρ0 , and pressure P . The depletion is a result of a large value of the magnetic field in the depressed region which corresponds to an increase in the magnetic pressure. The relative strength of the magnetic pressure to the thermodynamic pressure is seen in the plasma beta function βP (t, r, θ) in Fig. 5.38. We also plot the magnetic pressure Pmagnetic = b2 /2 along with the thermodynamic 186 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Model M3 v∞ 0.9 Γ 4/3 O(βP∞ ) 2 c∞ s 0.1 Table 5.5: The parameters used for the system studied with an accreting magnetofluid with a magnetic field orientated parallel to the rotation axis of the black hole. The fourth column is the order of magnitude selected for the asymptotic plasma beta parameter. The hydrodynamic variables selected were chosen to allow for asymptotically supersonic flow. The last column indicates the asymptotic speed of sound. pressure, Pthermal , in Figs. 5.39, 5.42 and total pressure, Ptotal , in Figs. 5.40, 5.43. The material in the depleted region is being pushed out by the magnetic pressure. This type of phenomenon is also seen in the region around the earth’s magnetopause where the magnetic pressure builds and the plasma drains out of the region along the magnetic field lines [179, 180]. Further research is necessary to assert that the density is draining along the field lines. To ensure that the depletion is a feature of the fluid flow and not a result of the specific resolution, we perform a convergence test focusing on a slice along r = 2. The results of these tests are seen in Fig. 5.46, where as the resolution increases the depletion becomes more and more apparent. The resulting accretion rates are seen in Figs. 5.49 and 5.50. Based on the magnetic energy accretion rates, we see that the magnetic field has not reached a steady state at the time that the evolution is terminated. The evolution of the hydrodynamic variables clearly have reached a steady state. Time evolution of the baryon rest mass density for model M1 and M2 may be found in the appendix in Figs. A.17 and A.18, respectively. 5.4.2 Magnetized Axisymmetric Accretion: a = 0 We simulate magnetized axisymmetric accretion onto an a = 0.5 black hole. A snapshot of model M3 at t = 2500M is seen in Fig. 5.51. We have a similar morphology as seen in the evolution of model M1. Model M3 has an interesting dynamical feature in that βP (t, r, θ) shows oscillations along the shock front in the tail. The oscillations propagate away from the black hole out of the domain. Time evolution of the density for model M3 is found in Fig. A.19. 187 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.38: The relativistic Bondi–Hoyle accretion simulation for model M1. The black hole is travelling parallel to the background magnetic field direction as measured by an observer at infinity. Top is the logarithm of the density, ρ0 (2500, r, θ), and the bottom is the log plot of the plasma beta function βP (2500, r, θ) profile. We see evidence that the magnetic field piles up on the downstream side of the black hole. The βP (t, r, θ) profile indicates that in the downstream side, near the black hole, the ideal MHD conditions may be breaking down, as the magnitude drops below unity. Note the high βP values in the tail of the wake, they indicate regions where the magnetic field drops significantly. 188 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.39: The relativistic Bondi–Hoyle accretion simulation for model M1. The black hole is travelling parallel to the background magnetic field direction as measured by an observer at infinity. The top is the thermodynamic pressure, Pthermal (2500, r, θ), the bottom is the magnetic pressure, Pmagnetic (2500, r, θ). We see that the thermal pressure decreases significantly on the axis of symmetry near the black hole on the downstream side, exactly where the magnetic pressure is at its largest. The total pressure is presented in Fig. 5.40. 189 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.40: The total pressure, Ptotal (2500, r, θ) (top), for the relativistic Bondi–Hoyle accretion simulation, model M1, and the hydrodynamic pressure for model M1 with |B| = 0 (bottom) at t = 2500. We see that the total pressure appears to have a similar profile to the hydrodynamic pressure in the flows studied in FI981, where a magnetic field is not considered. 190 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.41: The relativistic Bondi–Hoyle accretion simulation for model M2 with a = 0. The black hole is travelling parallel to the magnetic field direction as measured by an observer at infinity. Top is the rest mass energy density profile, and the bottom is the plasma beta βP profile. We see evidence that the magnetic field piles up on the downstream side of the black hole. The βP profile indicates in the downstream side, the near the black hole, the ideal MHD conditions may be breaking down, as the magnitude drops below unity. Note the high βP regions in the tail of the wake, which indicate regions where the magnetic field drops significantly. These regions start near the black hole and propagate out of the domain along the shock. 191 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.42: The relativistic Bondi–Hoyle accretion simulation for model M2. The black hole is travelling parallel to the background magnetic field direction as measured by an observer at infinity. The top is the thermodynamic pressure, Pthermal (1400, r, θ), the bottom is the magnetic pressure, Pmagnetic (1400, r, θ). We see that the thermal pressure decreases significantly on the axis of symmetry near the black hole on the downstream side, where the magnetic pressure on the downstream side of the black hole is at its largest. These results are consistent with those of model M1. The total pressure is presented in Fig. 5.40. 192 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.43: The total pressure, Ptotal (1400, r, θ) (top), for the relativistic Bondi–Hoyle accretion simulation, model M2 and the hydrodynamic pressure for model M2 with |B| = 0 (bottom) at time t = 1400M . In agreement with the total pressure for model M1, we see that the total pressure on the downstream side of the black hole has a similar profile to the hydrodynamic profile, Fig. 5.45, and in the purely hydrodynamic study performed in FI981. 193 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.44: The relativistic Bondi–Hoyle accretion pressure cross-section for model M1 with a = 0. We have plotted the three pressures on the event horizon at t = 2500: thermal, Pthermal (2500, 2, θ); magnetic, Pmagnetic (2500, 2, θ); and total, Ptotal (2500, 2, θ). We see that the magnetic pressure is at its greatest on the axis of symmetry and balances the thermodynamic pressure near θ ∼ 0.4, while the purely hydrodynamic model (|B| = 0) has a pressure maximum on the axis of symmetry. 194 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.45: The relativistic Bondi–Hoyle accretion pressure profiles for model M1 with a = 0. We have plotted the three pressures, thermal, Pthermal (2500, r, θ), magnetic, Pmagnetic (2500, r, θ), and total, Ptotal (2500, r, θ), for both the upstream region, θ = π, (left) and the downstream region, θ = 0 (right). In these plots we show the radial domain from the event horizon to the outer domain at rmax = 50. When comparing to the purely hydrodynamic model, we see that upstream the thermal pressure matches the hydrodynamic pressures; however in the downstream region the total pressure is a closer match to the hydrodynamic pressure. The downstream pressure profile for model M1 does have a larger maximum pressure than the hydrodynamic model allows. 195 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.46: A convergence test for the relativistic Bondi–Hoyle accretion simulation for model M1. We see that the baryon rest mass density, ρ0 , is converging to a shock front at θ ∼ 0.9. The density profile along the axis in this simulation indicates that the fluid is evacuating the axial location. This is in contrast to the simulations performed in FI981, where the axis marked a density maximum. Level 1 uses (Nr , Nθ ) = [200 × 80], level 2 (Nr , Nθ ) = [400 × 160], and level 3 (Nr , Nθ ) = [800 × 320]. 196 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.47: The relativistic Bondi–Hoyle accretion pressure cross section for model M2 with a = 0. We have plotted the three pressures, thermal, Pthermal (1400, 2, θ), magnetic, Pmagnetic (1400, 2, θ), and total, Ptotal (1400, 2, θ), along the event horizon at t = 1400. We see that the magnetic pressure is at its greatest on the axis of symmetry and balances the thermodynamic pressure near θ ∼ 0.6, while, like model M1, the hydrodynamic pressure is maximal on the axis of symmetry. We also see that the presence of a magnetic field allows for a wider opening angle of the Mach cone. 197 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.48: The relativistic Bondi–Hoyle accretion pressure profiles for model M2 with a = 0. We have plotted the three pressures, thermal, Pthermal (1400, r, θ), magnetic, Pmagnetic (1400, r, θ), and total, Ptotal (1400, r, θ), in both the upstream θ = π (left) and downstream θ = 0 (right) region. The upstream region shows a close agreement between the thermal pressure and the hydrodynamic pressure; however, it is apparent that the two models are not in as close agreement as seen in Fig. 5.45 for model M1. The downstream pressure profiles for the hydrodynamic model and model M2 are in close agreement. Model M2 allows for a larger downstream pressure maximum than the purely hydrodynamic model allows. 198 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.49: The normalized mass accretion for models M1 and M2. The flow with larger Γ accretes more mass, which is in agreement with the hydrodynamic systems studied in FI981 [1]. We have also plotted the mass accretion rates with no magnetic field present. The presence of the magnetic field has a marginal affect on the mass accretion rate. 199 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.50: The relativistic Bondi–Hoyle energy accretion for models M1 and M2 with the corresponding simulations with no magnetic field present. The magnetic field energy accretion rate, E˙ M (bottom), has not yet settled in the time shown here, but the total energy, E˙ (top), is clearly steady, which indicates that the hydrodynamic energy contribution in this simulation dominate the system. The trend shown in the total energy accretion rate is in agreement with the hydrodynamic systems studied in Font et al.[1]. The oscillations in the magnetic energy accretion rate decrease as a function of resolution. 200 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.51: Magnetohydrodynamic pressure profiles in the upstream (left) and downstream (right) regions for model M3. We see that the thermal pressure in the upstream region of the black hole is small by comparison to the pressure in the wake in the downstream region. The image on the left shows that there is an increase in pressure as the fluid reaches the black hole, as is expected from Bondi accretion [1]. The bulk of the pressure piles up in the wake of the black hole, and the thermal pressure begins to decrease near the black hole in the wake, while the magnetic pressure reaches a maxima on this slice. The above slices are recorded at time t = 2500M , and has not yet reached a steady state. As with model M1 the upstream pressure profile closely matches the thermal pressure profile, while the downstream pressure profile closely matches the total pressure. In this configuration, the hydrodynamic pressure profile has a larger maximum than the total pressure profile in model M3. 201 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.52: The relativistic Bondi–Hoyle accretion simulation for model M3 with a = 0.5. The black hole is travelling parallel to the magnetic field direction as measured by an observer at infinity. Top is the pressure profile, and the bottom is the plasma beta βP profile. We see evidence that the magnetic field piles up on the downstream side of the black hole. The βP profile indicates that in the downstream side, near the black hole, the ideal MHD conditions may be breaking down, as the magnitude drops below unity. Note the high βP nodes in the tail of the wake, they indicate regions where the magnetic field drops significantly. These regions start near the black hole and propagate out of the domain along the shock front. 202 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.53: The relativistic Bondi–Hoyle accretion simulation for model M3. The black hole is travelling parallel to the background magnetic field direction as measured by an observer at infinity. The top is the thermodynamic pressure, Pthermal (2500, r, θ), while the bottom is the magnetic pressure, Pmagnetic (2500, r, θ). We see that the thermal pressure decreases on the axis of symmetry near the black hole on the downstream side, where the magnetic pressure on the downstream side of the black hole is at its largest. These results are consistent with those of model M1. The total pressure is presented in Fig. 5.54. 203 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.54: The total pressure, Ptotal (2500, r, θ) (top), for the relativistic Bondi–Hoyle accretion simulation, model M3 and the hydrodynamic pressure for model M3 with |B| = 0 (bottom) at time t = 2500M . As with the total pressure for models M1 and M2, we see that the total pressure on the downstream side of the black hole takes on a similar profile as the hydrodynamic pressure profile. 204 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.55: The L2 -norm of the auxiliary function Ψ(t, r, θ) for the relativistic Bondi–Hoyle accretion simulation model M3. This trend shows a small increase of the auxiliary function Ψ, which indicates that there is an increase in the ∇ · B = 0 violations. However, with higher resolution we see that the ∇ · B = 0 violations are reducing to zero. The slow increase of the constraint violation over time may account for the accretion of magnetic energy not reaching a steady state. In Figs. 5.58, 5.59, and 5.60 we see the accretion rates for the magnetized relativistic Bondi– Hoyle accretion flow, model M3. In these figures, the asymptotic magnetic field is parallel to the trajectory of the black hole. The magnetic field flux is embedded in the accreting fluid, and consequently also builds in the tail shock. The build-up of the magnetic field strength leads to a decrease in the accretion rate for matter onto the black hole, which is seen in the accretion plots in Figs. 5.59 and 5.60. 205 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.56: A convergence test for the relativistic Bondi–Hoyle accretion simulation for model M3 with a = 0.5 at time t = 300M . The cross section is taken at r = 2M We see that the system is converging to a shock front at θ ∼ 0.8. 206 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.57: Pressure cross-section at r = 2M for model M3 with O(β) = 2, and |B| = 0. We see that the cross-section is completely different between the two models. The hydrodynamic model does not experience a depletion region, and we see that with a rotating black hole the pressure in the purely hydrodynamic model is different than exhibited in model M3, including a smaller Mach cone opening angle. 207 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.58: The relativistic Bondi–Hoyle energy accretion for model M3. The magnetic field has not settled in the time shown here, but the total energy is clearly steady, which indicates that the hydrodynamic energy contribution in this simulation dominates the system. 208 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.59: The relativistic Bondi–Hoyle accretion simulation for model M3 with a = 0.5. The magnetic field has not settled in the time shown here, but the hydrodynamic variables are clearly steady. We present the radial momentum accretion rate with and without the presence of the magnetic field. It is evident that the magnetic field does not have a large impact on the momentum accretion rate. 209 5.4. MAGNETOHYDRODYNAMIC BONDI–HOYLE ACCRETION ONTO A BLACK HOLE Figure 5.60: The relativistic Bondi–Hoyle accretion simulation for model M3 with a = 0.5. The magnetic field has not settled in the time shown here, but the hydrodynamic variables are clearly steady. In the top plot we present the mass accretion rate with and without a magnetic field. There is a slight difference between the accretion rates towards the end of the simulation. This is attributed to the evolution of the magnetic field. In the bottom plot we present the difference between the angular momentum accretion with and without the presence of the magnetic field. We see that a net azimuthal accretion rate is oscillatory. During the transition period there is a large spike in the angular momentum accretion. This is attributed to the initialization of the fluid variables to their asymptotic uniform values and is deemed a numerical artifact. 210 CHAPTER 6 CONCLUSIONS AND FUTURE DIRECTIONS 6.1 Conclusions Our studies revealed several interesting new results for the Bondi–Hoyle accretion problem when using different models for the background fluid. We recapitulate these results below and draw our conclusions. 6.1.1 Ultrarelativistic Hydrodynamics For our investigation of the ultrarelativistic hydrodynamic Bondi–Hoyle accretion problem we conclude that all flows of this configuration and set of boundary conditions are stable for long-term evolution. In the axisymmetric UHD accretion problem, all flows develop a steady state after a short transition period and maintain stability well after the steady state is established. Unlike the original study by Font et al. [1, 2] that used rmax = 50, this simulation for the models presented required at a minimum rmax = 100 to capture the evolution without interference by domain boundaries. A parameter survey where we test the functional dependence of the accretion rate on the value of rmax is a future direction for this project. Only preliminary dependence has been investigated so far, and there is evidence that the domain does impact the accretion rate for our system. Font et al.used the usual hydrodynamic equations of motion (Eqn. (2.107) using (2.117), (2.118) and (2.119) with B i = 0) to study fluids with speeds of sound in the ultra relativistic regime, √ v∞ > Γ − 1. Our study focused on a fluid with an internal energy sufficiently large that we are able to neglect the conservation of the baryon rest mass density entirely, therein solving a different set of hydrodynamic equations. The studies performed by Font et al.set the outer radial boundary at rmax = 50, and assumed that this is sufficiently far from the black hole to prevent numerical artifacts from disrupting the calculations even for their ultrarelativistic study. Our studies of the ultrarelativistic regime show that using rmax = 50 results in a stable accretion flow that is 211 6.1. CONCLUSIONS marred with numerical artifacts. Our results show that the domain must be carefully monitored to prevent artificial numerical results. Furthermore, when studying the axisymmetric evolution of our formulations of the ultrarelativistic hydrodynamic system we discovered that marginally supersonic flows exhibit a detached shock. The detached shock formation was not achieved in previous simulations. However, as stated in Chap. 5 we require further studies to determine if the bow shock is a result of choosing a different parameter regime, the choice of the domain of integration, or if it is a direct result of the choice of physical fluid model. We leave answering this question to a future investigation. When larger asymptotic velocities were studied, the flow morphology changed dramatically. The bow shock did not form, instead the tail shock that forms early in the simulation remained attached to the black hole. This is in agreement with previous studies presented in [1]. The infinitely thin-disk approximation has a different evolution. The flow, even at large velocities, produces an outward travelling shock wave, propagating upstream. To study this problem effectively we required the use of a much larger domain than those used in the axisymmetric study. In the simulations where the domain size was small, the shock wave would quickly reach the upstream boundary, and alter the boundary conditions in the upstream boundary. The leading shock was found to behave similar to the bow shock found in the axisymmetric v∞ = 0.6 evolution. A tail shock developed, and remained attached to the black hole. The accretion rates were stable and steady in these simulations when a larger domain was used. 6.1.2 Magnetohydrodynamic Accretion The magnetized fluid background configuration used in our study did not result in a steady accretion flow. We performed a preliminary parameter survey by studying a small range of plasma beta parameters, and found that only larger plasma beta parameters would reach a steady state. We were restricted to plasma beta parameters above unity due to the numerical methods used, as discussed in Chaps. 2 and 3. Considering larger field strengths requires the development of new methods to handle modified equations of motion. Although the flows did not reach a steady state, we can conclude that the flows up to time of termination were stable. Ultimately, there is a question of how we should maintain the magnetic field constraint. We used the diffusive hyperbolic divergence cleaning method which allowed the violations of the nodivergence constraint to be treated as a flux for an auxiliary field, which we could quench as 212 6.2. SUMMARY necessary. In this study we considered a quenching parameter of cp = 12, which is used for strong shock conditions [122], and used ch = 1, such that the violations will propagate out of the domain as fast as possible, without the need to decrease the Courant factor to keep the results meaningful. We investigated both the total field contributions, as well as the magnetic field contributions. The total field accretion rates achieve a steady rate; however, the magnetic field contributions were relatively small, and did not reach equilibrium. The simulations were pushed to long time evolution, and a steady state was not achieved. Further research considering larger asymptotic magnetic fields will be necessary. The morphology of the ideal MHD accretion shows a rather significant impact by the presence of a magnetic field. The downstream axial location shows evidence of a depleting density, whereas in the hydrodynamic models presented by Font et al. [1], this region has a maximum density. When we studied the accretion onto a rotating black hole, the profile of βP (t, r, θ) indicated that the magnetic field experienced deficits along the shock front along the tail which propagate out of the domain. During the simulations, the auxiliary field, Ψ, remained well below unity, and by checking the convergence of this field we saw that it tends to zero, but would slowly grow as the evolution proceeded. During the simulations presented here the constraint violations did not grow sufficiently large to disrupt the system. The fluid rest mass density and thermal pressure evacuate the region immediately downstream of the black hole. The depleted region corresponds exactly with the region where the magnetic field (and magnetic pressure) is at its largest. The large magnetic pressure leads us to consider that the fluid is being evacuated in a similar fashion to the plasma depletion layer near the Earth’s magnetopause. More research is required to know for certain that this is the same effect. 6.2 Summary The key results from our investigation are: • UHD systems are highly sensitive to the location of the boundary of the domain of integration; • UHD axisymmetric systems are capable of producing a bow shock; • UHD non-axisymmetric infinitely thin-disk models produce both bow shocks and tail shocks; 213 6.3. FUTURE DIRECTIONS • Both of the UHD configurations studied are stable; • MHD axisymmetric systems produce a density depletion region immediately downstream of the black hole; • Although the MHD flows did not reach a steady state, the flows were stable; • A new field, Ψ, may be used to monitor the convergence rates of MHD systems. These results were not previously known, and thus are not found in the literature. We now discuss the future directions that numerical codes can be modified to investigate. 6.3 Future Directions From our research alone there are several open questions, the first of which is the combination of the ultrarelativistic evolution with the presence of a magnetic field. This would be a useful simulation to determine the accretion rates for early Universe black holes, where the surrounding fluid is thought to be sufficiently hot to justify the ultrarelativistic approximation, while at the same time we expect the presence of a weak background magnetic field [181]. The codes developed for the UHD simulations are sufficiently generalized to also allow for the evolution of a background magnetic field. In future studies we also expect to analyze the magnetic systems presented here in three dimensions. Extending the study to three dimensions will allow us to study any configuration of the black hole relative to the asymptotic magnetic field. Another open question from our line of research is the dependence the accretion rates have on the value of rmax , the maximum radial extend of our domain. Our studies show that rmax has an impact on the accretion rates, and on the flow morphology. The use of ideal magnetohydrodynamics limited our study to cases where the magnetic field strength was insufficient to significantly affect the accretion rates. If we extend our numerical method to allow for a transition to force-free electrodynamics (FFED) [127], in particular near the black hole, we will be able to increase the asymptotic magnetic field strength and study the strong field limit. This is important when investigating binary mergers where the magnetic field emanates from the neutron star but extends beyond the matter into the surrounding spacetime where there is only vacuum. 214 6.3. FUTURE DIRECTIONS Further, there has been some unpublished development of codes that allow hydrodynamic codes to use a combination of an exact Riemann solver with the approximate Riemann solvers. This allows the code to capture a true vacuum, while still maintaining the computational efficiency of the approximate solvers. Implementing such effects in an MHD code will aid in the understanding of true (magneto)fluid dynamics. By modifying the standard approximate Riemann solvers to include the limit where a physical vacuum is expected allows us to properly model thick accretion disks surrounded by true vacuum. Research into relativistic plasma astrophysics is still in its infancy. With the common availability of high powered supercomputers and the persistent study of higher accuracy approximate solutions to the (magneto)hydrodynamic equations of motion, we are able to probe closer and closer to the true behaviour of fluid systems. Only in the last decade have the numerical tools used to study relativistic magnetized fluid systems been developed. Because of this we have only scratched the surface of all the different effects and possible configurations that observation may one day detect. We list a few of the extensions that we find to be the most interesting. This is not a complete list of all the possibilities, but does offer a sample of how much more there is to do in the field of fluid dynamics and magnetohydrodynamics. Accretion onto Neutron Stars The research presented studied accretion onto a black hole. In future studies, we expect to extend this research towards modelling accretion onto other astrophysical compact objects such as neutron stars. To study these objects we will need to make large changes to the existing code, including the likely possibility that the compact object itself has a magnetic field, such is the case with pulsars. To achieve these goals the code may need to be extended to include the following physical fields/effects. Force-Free Electrodynamics When the magnetic energy is much greater than the internal hydrodynamic energy i.e. b2 ≫ ρ we enter a physical regime where the hydrodynamic contributions are negligible and we need to consider force free electrodynamics, wherein one uses the force free idea described above and apply it to the electrodynamic equations [182, 156]. This reduces the system to the study of the full complement of the electrodynamics equations and the 3-momentum/3-velocity fields. Such a 215 6.3. FUTURE DIRECTIONS system is necessary when the plasma beta parameter is much less than unity, so the magnetic field energy and magnetic pressure dominate the hydrodynamic quantities. If at any point an ideal MHD system develops a plasma beta that drops well below unity, the ideal MHD conditions break down and a new set of equations of motion are necessary. An example of this occurring is near the event horizon of a rotating black hole, inside the ergo-sphere, where the frame dragging effects of the black hole wind the magnetic field lines increasing their local field strength. Such an effect has been studied using the full force-free approximation by Komissarov [171, 56, 183]. He showed that accounting for an electric field arising from the frame dragging is important for the behaviour of accreting material, but in his studies, he did not include the dynamics of matter away from the black hole. By allowing the force-free approximation to be embedded in a GRMHD code, such that as the ideal MHD approximation is violated a new set of equations are automatically accounted for, we could get a better picture of the accretion process onto a moving black hole. Radiation Hydrodynamics I intend to continue my study of hydrodynamic systems by investigating a more general case of heat transport beyond the idealized situations. Some research into thermodynamic effects has been studied by investigating thermodynamic radiation from a Stephan-Boltzmann law for blackbody radiation. This is a necessary process to study if one is in the lower energy limit. If we deviate too far from the IR spectrum we usually have to leave the blackbody treatments as a mild effect relative to other more dominating radiation effects [184, 185]. More recent work on this effort are considered in Romatschke [186, 187]. By including the radiative effects we expect to model the observable energy signatures of accreting matter, and will work towards obtaining a more complete picture of the accretion process for other astrophysical compact objects. Full Electromagnetic hydrodynamics Recently, the study of MHD systems has given us the tools needed to study the full electromagnetic hydrodynamic systems. In these systems conductance is no longer infinite, and consequently electric fields are not longer explained using the ideal MHD condition. Some of the leading authors on this material are Palenzuela [124], and Komissarov [182]. Each group have formulations for the full electromagnetic hydrodynamics still in the ideal fluid limit, and expressed for special relativistic metrics. It would be of great interest to study the effects of the full system in general relativity 216 6.3. FUTURE DIRECTIONS even with stationary metrics. Palenzuela et al. have shown in [188, 189], the effects if we were to include the presence of both an electric and magnetic field as the neutron stars merge. Farris et al. [72] studied the relativistic Bondi and Bondi–Hoyle accretion during a binary merger of two neutron stars, where he also calculated the electromagnetic signal radiating from the merger by assuming the hydrodynamic background contained a weak magnetic field which was dynamically unimportant. By following the work of Palenzuela et al. we would need to repeat the work performed by Farris using a full electromagnetic hydrodynamic treatment of the fluid background, giving us a better picture of the electromagnetic signal from the merger. Use of this formulation would also improve our model for the accretion disks that have domains that extend into the Kerr hole ergosphere. It is thought that the black hole will induce an electric field for magnetized material that enters the ergosphere [172, 171, 56, 183]. Study of how this field interacts with the surrounding plasma will be essential in the study of extremal black holes. Equation of State Finally, there has been an effort by some members of the MHD community to attempt to study more “realistic” equations of state. The range of possible equations of state is quite large and it is a difficult task to review all expressions here. Most of the research in (magneto)hydrodynamic flows performed to date uses the ideal gas equation of state as a means to use a well known and well understood equation of state to determine what kind of dynamics ensue. Adhering so closely to such an approximation is limiting. In some cases the idealization is used for the ease of closed form calculations or for ease of numerical computation. An example of a more general equation of state was studied by Synge to describe what is now known as a “Synge gas” [190]. For consistency with the kinetic theory of gases, in the limit of negligible mean free path, Taub determined that we cannot treat the specific heat ratio as a constant [114]. With modern computers and algorithms, this type of generalization, although more expensive that the current approximations, is worth pursuing. A recent example of this pursuit may be found in [115]. When we model accretion onto a neutron star, the use of an accurate equation of state for the accreting matter will be beneficial. By studying accretion onto black holes we get a sense of what to expect when we study accretion onto other astrophysical objects. Ultimately the study of accretion onto compact objects other than black holes will have computational complexities and different physical properties that will require the development of a set of general numerical routines 217 6.3. FUTURE DIRECTIONS that may include any or all of the above proposed extensions. 218 BIBLIOGRAPHY [1] J.A. Font and J.M. Ib´ an ˜ez. A Numerical Study of Relativistic Bondi–Hoyle Accretion onto a Moving Black Hole: Axisymmetric Computations in a Schwarzschild Background. The Astrophysical Journal, 494:297–316, (1998). [2] J.A. Font and J.M. Ib´ an ˜ez. Non-axisymmetric Relativistic Bondi–Hoyle Accretion onto a Schwarzschild Black Hole. MNRAS, 298:835–846, (1998). [3] J.A. Font, J.M. Ib´ an ˜ez, and P. 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Stone, and B.E. Wood. The Dynamic Heliosphere: Outstanding Issues. Space Sci. Rev. , 143:57–83, (2009). [180] Y.L. Wang, J. Raeder, and C.T. Russell. Plasma depletion layer: Magnetosheath flow structure and forces. Annales Geophysicae, 22:10011017, 2004. [181] D. Grasso and H. Rubinstein. Magnetic Fields in the Early Universe. Physics Reports, 348:163–266, (2001). [182] S.S. Komissarov and M.V. Barkov. Activation of the Blandford-Znajek Mechanism in Collapsing Stars. MNRAS, 397:1153–1168, (2009). [183] S.S. Komissarov and J.C. McKinney. The ’Meissner Effect’ and the Blandford–Znajek Mechanism in Conductive Black Hole Magnetospheres. MNRAS, 377:L49–L53, (2007). [184] G.S Denicol, T. Kodama, T. Koide, and P. Mota. Causal Theory of Relativistic Dissipative Hydrodynamics. Braz. J. Phys., 37, (2007). [185] G.S Denicol, T. Kodama, T. Koide, and P. Mota. Causality, Memory Effect, and Relativistic Dissipative Hydrodynamics. Braz. J. Phys., 37, (2007). [186] P. Romatschke. New arxiv:0902.3663v3, (2009). Developments in Relativistic Viscous Hydrodynamics. [187] P. Romatschke. Relativistic Viscous Fluid Dynamics and Non-Equilibrium Entropy. arxiv:0906.4787v1, (2009). [188] C. Palenzuela, L. Lehner, and S. Liebling. Dual Jets from Binary Black Holes. Science, 329(5994):927–930, (2010). [189] C. Palenzuela, L. Lehner, and S. Yoshida. Understanding Possible Electromagnetic Counterparts to Loud Gravitational Wave Events: Binary Black Hole Effects on Electromagnetic Fields. Phys.Rev.D, 81:084007, (2010). [190] J.L. Synge. The Relativistic Gas. (1957). [191] LORENE: Langage Objet pour la RElativit´e Num´eriquE. http://www.lorene.obspm.fr/. [192] F. Pretorius. PAMR Reference Manual. 228 APPENDIX A TIME EVOLUTION In this appendix we present examples of the time evolution of the pressure in the relativistic Bondi-Hoyle accretion simulations studied in this thesis. Specifically we present: 1. A.3 presents axisymmetric UHD accretion on an a = 0 black hole for model U2; 2. A.4 presents axisymmetric UHD accretion on an a = 0 black hole for model U4; 3. A.5 presents axisymmetric UHD accretion on an a = 0.5 black hole for model U7; 4. A.7 presents axisymmetric UHD accretion on an a = 0.99 black hole for model U11; 5. A.11 presents thin-disk UHD accretion on an a = 0 black hole for model U12, rmax = 50; 6. A.13 presents thin-disk UHD accretion on an a = 0 black hole for model U13, rmax = 50; 7. A.14 presents thin-disk UHD accretion on an a = 0 black hole for model U12, rmax = 1000; 8. A.15 presents thin-disk UHD accretion on an a = 0 black hole for model U13, rmax = 1000; 9. A.17 presents axisymmetric MHD accretion on an a = 0 black hole for model M1; 10. A.18 presents axisymmetric MHD accretion on an a = 0 black hole for model M2; 11. A.19 presents axisymmetric MHD accretion on an a = 0.5 black hole for model M3. We also present one image which shows the shock-capturing properties of the UHD code we developed in Fig. A.16. The visualization was produced using a program called “Data Vault” created by F. Pretorius, M. Choptuik, M. Snajdr, and R. Stevenson. One of the features of this program is that it automatically selects the scale for each time frame based on the maximum and minimum of the data set at each timestep. Thus one needs to be careful in comparing the figures at different time steps as the same colours may represent very different values for the pressure. In all cases red corresponds 229 A.1. AXISYMMETRIC ULTRARELATIVISTIC FLOW zˆ r θ Figure A.1: The axisymmetric accretion setup. The flow enters from the domain boundary in the “upstream region” along π/2 ≤ θ ≤ π and flows up past the black hole represented by the black semi-circle in the middle of the domain. The fluid then enters the downstream region and, unless flowing into the black hole will proceed to the downstream outer domain along 0 ≤ θ < π/2 where it flows out of the domain. to the minimum pressure while blue corresponds to the maximum pressure as designated on each timestep. A.1 Axisymmetric Ultrarelativistic Flow In this section we present snapshots of the axisymmetric ultrarelativistic fluid flow. Recall that in axisymmetry the fluid is confined to flow in the r and θ directions only. The symmetry assumes that the flow is identical for any coordinate φ selected. The following figures show the time evolution of the fluid pressure. 230 A.1. AXISYMMETRIC ULTRARELATIVISTIC FLOW tail shock black hole bow shock Figure A.2: A diagram containing the major features present in the following flow profiles. On the left we present the flow with a bow shock, while on the right we present a flow with a tail shock. 231 A.1. AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.3: Time evolution of P (t, r, θ) for model U2 with a = 0. We see the evolution of the ultrarelativistic fluid as it flows past the black hole. The formation of the a tail shock is evident in the first two frames. This shock expands quickly and eventually detaches from the black hole by the 4th frame. In the last two frames we see the detached shock flow upstream. 232 A.1. AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.4: Time evolution of P (t, r, θ) for model U4 with a = 0. We see the evolution of the ultrarelativistic fluid as it flows past the black hole. The formation of the a tail shock is evident in the first two frames. This shock begins to expand, but does not break contact with the black hole. The last several frames show that the fluid flow is steady. 233 A.1. AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.5: Time evolution of P (t, r, θ) for model U7 with a = 0.5. We see the evolution of the ultrarelativistic fluid as it flows past the rotating black hole does not change morphology significantly when compared to the a = 0 black hole. Once again we have a tail shock and a steady flow. 234 A.1. AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.6: Longer time evolution for model U7 with a = 0.5. 235 A.1. AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.7: Time evolution of P (t, r, θ) for model U11 with a = 0.99. As with other models the tail shock forms; however, here we see that the shock attaches to the black hole much closer to the upstream side. 236 A.1. AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.8: Longer time evolution of P (t, r, θ) for model U11 with a = 0.99. The flow reaches a steady state. 237 A.2. NON-AXISYMMETRIC ULTRARELATIVISTIC FLOW yˆ r φ x ˆ Figure A.9: The non-axisymmetric infinitely thin disk accretion setup. The material enters the domain along the π/2 ≤ φ ≤ 3π/2 boundary, known as the upstream region. The fluid then flows “up” the page, past the black region in the diagram, denoting the black hole, to the downstream region where if it makes contact with the domain on −π/2 < φ < π/2 the fluid leaves the domain of integration. We refer the reader to Fig. A.10 for a diagram of the shocks found in the flow morphology. A.2 Non-axisymmetric Ultrarelativistic Flow In this section we present snapshots of the non-axisymmetric infinitely-thin disk ultrarelativistic fluid flow. Recall that for this model the fluid is confined to flow in the r and φ directions only. We confine the flow to the equatorial plane. The following figures present the time evolution of the fluid pressure. 238 A.2. NON-AXISYMMETRIC ULTRARELATIVISTIC FLOW tail shock black hole bow shock Figure A.10: The flow morphology for a typical simulation using the infinitely thin-disk approximation. We have clearly labelled the bow shock, black hole, and the tail shock in this diagram. 239 A.2. NON-AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.11: Time evolution of the infinitely thin-disk accretion of P (t, r, φ) for model U12 with a = 0. We plot the logarithm of the pressure profile here; other scalar fields have a similar profile. The tail shock is clearly visible. 240 A.2. NON-AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.12: Time evolution of the thin-disk accretion of P (t, r, φ) for model U12 with a = 0. We plot the pressure profile here; other scalar fields look similar. The last several frames shows that the evolution has become steady state. 241 A.2. NON-AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.13: Time evolution of the thin-disk accretion if P (t, r, φ) for model U13 with a = 0. When the flow is faster the width of the tail shock decreases. Notice the wave impinging on the upstream boundary in the second last frame. When this makes contact with the boundary we expect an amplification of the boundary effects to enter the simulation. 242 A.2. NON-AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.14: Time evolution of the thin-disk accretion of P (t, r, φ) for model U12 with rmax = 1000 and a = 0. We plot the logarithm of the pressure profile here, other scalar fields have a similar profile. The tail shock is clearly visible. 243 A.2. NON-AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.15: Time evolution of the thin-disk accretion of P (t, r, φ) for model U13 with rmax = 1000 and a = 0. When the flow is faster the width of the tail shock decreases. 244 A.2. NON-AXISYMMETRIC ULTRARELATIVISTIC FLOW Figure A.16: A snapshot of the pressure for model U13 at time t = 500M . In this image the black hole is travelling diagonally across the page. This image clearly shows that the bow shock moving upstream is sharply defined even with a low resolution ∆r = 0.25. 245 A.3. AXISYMMETRIC MAGNETOHYDRODYNAMIC FLOW A.3 Axisymmetric Magnetohydrodynamic Flow In this section we present snapshots of the axisymmetric magnetohydrodynamic relativistic fluid flow. In this model we again confine the fluid to flow in the r and θ directions only. The geometric setup is identical to that in Fig. A.1. The following figures present the time evolution of the fluid rest mass density. 246 A.3. AXISYMMETRIC MAGNETOHYDRODYNAMIC FLOW Figure A.17: Time evolution of the axisymmetric of ρ0 (t, r, θ) accretion for model M1 with a = 0. We plot the log of the density profile here. The last several frames shows that the density evolution has reached a steady state. 247 A.3. AXISYMMETRIC MAGNETOHYDRODYNAMIC FLOW Figure A.18: Time evolution of the axisymmetric accretion of ρ0 (t, r, θ) for model M2 with a = 0. We plot the log of the rest mass density profile here. The last several frames shows that the rest mass density evolution has not reached a steady state in the time presented. 248 A.3. AXISYMMETRIC MAGNETOHYDRODYNAMIC FLOW Figure A.19: Time evolution of the axisymmetric accretion of ρ0 (t, r, θ) for model M3 with a = 0.5. We plot the log of the density profile here. The last several frames shows that the density evolution has reached a steady state. 249 APPENDIX B CODE DEVELOPMENT A major part of the research presented in the thesis required that I wrote my own high-resolution shock-capturing code. Below are the implementation details as well as the difficulties, and technical issues that arose during production. In the following, just as with the thesis, the dimensionality described below refers to the number of spatial dimensions. An additional temporal dimension is always present. B.1 Stages The development of a full three dimensional code that solves fluid flow on a curved spacetime background was done in stages. The first stage, and likely the easiest, was to develop a one dimensional special relativistic code using Cartesian coordinates. This was done in a matter of weeks, and is a good exercise to develop a firm understanding of HRSC methods. Using this code one can quickly and easily test different reconstruction techniques and primitive variable solvers. The results of this code are seen in Chap. 4. Due to numerical efficiency of the approximate Riemann solvers, and the speed of modern processors I found that the (non-parallelized) one dimensional simulations could be performed using high resolution i.e. N ∼ 6400, while only taking approximately 30 minutes to run. It is difficult to see new physical information coming from one dimensional codes. However, they are useful in performing research on optimal approximate Riemann solvers and flux limiters. The Roe, HLL, and Lax–Friedrich flux limiters described in Sec. 3.8.3 are the most popular methods due to their low rates of failure, but this is not a complete list of flux approximations. For a larger list, I refer the reader to [21]. Likewise for the variable reconstruction techniques: minmod and MC as described in Sec. 3.8.2 are the simplest second order reconstruction techniques available. For a higher order reconstruction technique I refer the reader to [140]. Using 1D codes, researchers can easily determine the effectiveness of any given approximate Riemann solver or flux limiter, especially when compared to exact Riemann solvers 250 B.1. STAGES such as described in [52]. The next stage was to develop a two dimensional special relativistic magnetohydrodynamic code (SRMHD). Again, a simplifying step as it does not have a lot of complications, but does help to develop ones intuition for more complicated codes. Developing the two dimensional code proved to be critical in providing basic tests of the shock capturing and symmetry capturing codes such as the test presented in Sec. 4.3.4. There are also a number of open questions that could be addressed using a two dimensional Cartesian code. One such question might be: what method do I need to use to prove numerical convergence of a turbulent system. This stage proved to be an essential stepping-stone towards the production of the final GRMHD code. Extending a one dimensional code to two dimensions made extending this code to three spatial dimensions is much easier. As a side project I also developed a 2D Cartesian code using a non-relativistic MHD model. This was a check comparing the effects of relativity versus the effects of numerical errors. The development of the non-relativistic code was an extra step, and it was also used to check the implementation of the constraint handlers described in Sec. 2.6.1. There is still active research using non-relativistic fluid models. While this two dimensional non-relativistic code is not used directly in the thesis research, it has many potential applications. Like its relativistic counterpart, the two dimensional non-relativistic code is also easily extended to three dimensions. The next stage could have been to extend the above code to three dimensions; however, this was beyond the scope of my research. Instead I switched focus to introducing curvilinear coordinates into the two dimensional SRMHD code. It is too easy to write code that is specific to a particular coordinate system which, when extended to general coordinates, fails. By focusing on this step, additional errors or “bugs” may be eliminated in a systematic way, saving a tremendous amount of debugging later. Additionally I generalized the code sufficiently that I could use an arbitrary stationary spacetime background. This helped produce the general relativistic magnetohydrodynamic (GRMHD) code used for the final thesis production runs. In making the code as generic as possible, so I also wrote a routine to calculate the 3 + 1 variables, and the various determinants described in Chap. 2 directly from the known four dimensional metric. By doing this I knowingly reduced the efficiency of the numerical routines; however, this allows me the flexibility to implement any future stationary spacetime metric. There are only two known fixed spacetime backgrounds for black holes, there are several available for other stellar bodies such as boson and neutron stars29 . 29 For a code that is capable of producing fixed spacetime backgrounds for different types of neutron stars, I refer the reader to codes such as LORENE [191]. 251 B.2. PARALLELIZATION To test the GRMHD code, I reproduced the results of Font et al. [1, 2, 3], as well as reproducing both the Michel one dimensional stead-state accretion [25] described in Sec. 4.6.1 and the magnetized Michel accretion described in Sec. 4.6.2. The one dimensional tests were useful to rout out array indexing problems in the two dimensional code. Any errors due to poor array indexing would appear as asymmetries in the final result. The one dimensional tests were also useful for testing boundary conditions. Further tests could have included the accretion torus [28]; however, given the focus of my study, reproducing many different previous results was not essential and I relied on convergence tests and independent residual test described in Secs. 4.1 and 4.2 to prove the accuracy of my results. It should be noted that the independent residual tests were only performed up to shock formation. The use of an independent residual test that may proceed past the production of a shock is still an open research area. The extension of this code to three dimensions is being done, but is still in the testing stages. No results are included in the thesis. The codes discussed above (with the exception of the three dimensional code) were all originally written for use on a single processor. I initially avoided parallelization for ease of implementing the different routines needed to perform finite volume discretization with high-resolution shock-capturing. Once the methods and codes were working properly I turned my attention to parallelization. B.2 Parallelization To parallelize the code I had two choices, an MPI-type or an openMP type. Since I was not working with shared memory machines, I chose to use the MPI-type parallelization routines. For the given code, I could have easily performed the domain decomposition by direct MPI calls. However, keeping in mind the future use of more complicated algorithms such as adaptive mesh refinement (AMR) I chose to use an existing infrastructure, PAMR (parallel adaptive mesh refinement) [170]. While the code used for the thesis does not use AMR, there were several advantages to using the PAMR package, including check-point saving utilities30 , and the relative ease of domain decomposition. Some modifications to PAMR were necessary before I could use that package, including 30 Check-point saving refers to saving the variables at specified intervals in the event that the code crashes due to memory problems or server/mainframe failures. 252 B.3. MAIN ROUTINE re-writing the way that PAMR handled the periodic physical boundary conditions for the domain. PAMR was written for the explicit use of finite differencing techniques which use vertex centred grid points. The FVM I implemented used cell-centred grid points as described in Sec. 3.8. For details on how to use the PAMR routines I refer the reader to [192]. It should be noted that using parallelization, the simulations I did still took days to complete using on the order of 100 processors. It was later found that reducing the check-point frequency improved the speed of the simulations by approximately 20%. Originally, I chose to output the full system to file every 1000 time steps to obtain data that produced high quality movies. For the simulations where the movies were unnecessary we output once every 5000 time steps was sufficient for the stability of the computer clusters used. Having given an overview of the codes produced, I now discuss the individual routines developed. For specific details I refer the reader back to the main text where needed. B.3 Main Routine To facilitate the use of the PAMR infrastructure I wrote the main routine using the C programming language, while the remaining subroutines and functions were written using the more numerically optimized FORTRAN 7731 . In doing this I also had to write wrappers allowing the C routines to call FORTRAN routines. The main routine was designed to handle only basic functions. Reading input parameters, writing output/check-points, calling the PAMR utilities to perform the domain decomposition allowing parallelization, calling the routine “makestep” which performed the time advancing calculations, and calling utility routines that performed the calculations such as the accretion rates. The purpose of the main routine is to perform the memory allocation calling subroutines routines or functions to perform all of the detailed calculations. I used FORTRAN to write the routines that required the most numerical calculations. By parallelization through domain decomposition I was able to leave all the FORTRAN routines that I developed for single processor calculations, such as the functions used to calculate the accretion rates, unaltered. The only FORTRAN subroutine that was altered was the makestep routine discussed below. I will now give more detail about the various subroutines, all of which are written in FORTRAN. 31 At the time of writing there is still debate over which programming language is most optimal for scientific computing. 253 B.3. MAIN ROUTINE B.3.1 Initialize The first routine called by the main routine initialized the fluid variables. I used this routine to initialize all configurations studied. This routine was only called once per simulation. The initialization routine also initialized the spacetime variables I stored. Since I am interested in stationary spacetimes I stored the values of the lapse, shift, 3-metric, etc. to prevent repeated calculations during time evolution. My code was written for a general stationary spactime so I chose not to use the known functions for the lapse and shift as presented in Sec. 2.3, but rather calculated them given the full 4D metric as described in Sec. 2.2. By generalizing the code like this, I reduced the efficiency of the routine; however, since memory is abundant we recovered some of the potentially lost time by storing the values of the metric functions at every grid point, and cell interfaces and only performing the calculation once. B.3.2 Makestep This routine was designed to perform the numerical integration, using the Euler, and Runge–Kutta methods as outlined in Sec. 3.8.7. This routine called the update routine described below to perform the cell updates, then called the routine to update the ghost cells. Ultimately this routine acts like a master routine used to monitor all parallel processes, which meant that I also needed to adapt this routine to synchronize all parallel processes, before proceeding to the next temporal iteration. The PAMR infrastructure has a built-in routines to facilitate this part of the procedure. B.3.3 Update The update routine did the bulk of the work, this routine organized the variable reconstruction, flux calculations and the source contributions. Each of these calculations were handled by further subroutine calls. It is possible that leaving all calculations inside a single routine would aid the compiler in producing an optimized executable. Using the subroutine approach aided in the debugging and routine testing performed along the way. I used flux splitting to calculate the flux contribution from each dimension; ultimately, each of the routines described below were called once for each dimension calculated. 254 B.3. MAIN ROUTINE Reconstruct Variables The first routine called by update calculates the variables at the cell interfaces. As discussed in Sec. 3.8, finite volume methods use the average values at the cell centre. Using special functions described in Sec.3.8.2 I used the information from surrounding cell centres to interpolate the values at the cell interface. By creating a single function call I was able to implement many different interpolation schemes. I chose to implement the piecewise constant, and two piecewise linear schemes (minmod and MC). The way in which the reconstruction algorithm was written leaves it open to also include the piecewise parabolic method. For my implementation of the routine, I reconstruct the primitive variables at the cell interface. This is not a unique choice, as we could have equally chosen to reconstruct the conservative variables. However, based on conversations with other people in this field, the reconstruction of the primitive variables prevents unphysical negative pressure solutions. Conservative Variable Calculation Given the primitive variables at the cell centre and the cell interface I calculate the corresponding conservative variables. There is a simple closed form solution used to perform this calculation found in Eqns. (2.74), (2.75), (2.76), and (2.77). Once the conservative variables are calculated I proceed to calculate the flux and the source contributions to each cell. Flux Calculations I also wrote a single routine to calculate the flux through each cell interface. Given the values of each variable at the cell interface I was able to use several different flux approximations. The techniques used in the thesis were the Roe, HLL and LF methods as described in Sec. 3.8.3. There are other schemes available, and in future developments of the code I will include the Marquina method. This will make the code more robust and amenable to other physical problems where the flux calculations used in my code may introduce numerical artifacts as discussed in [144]. Other flux approximations not described here are outlined in [144]. The flux calculation routines used both the primitive and conservative variables at the cell interfaces. It is possible to only use the primitive variables for the flux calculations; however, this would make the code less readable by the user, and would inevitably lead to the production of coding errors. 255 B.3. MAIN ROUTINE The flux calculations I wrote required the use of LAPACK routines to calculate the eigenvalues and eigenvectors for the flux Jacobian described in Chap. 3. These routines were easily tested since the eigenvalues for the GRMHD system are known and described in Sec. 3.4. It was not clear if it was numerically faster to calculate the eigenvalues from the Jacobian or if it was faster to calculate them from the algebraic equations in Sec. 3.4, and then solving the quartic polynomial needed for the values of the fast and slow magnetosonic wave speeds. Ultimately this is open for testing and may be a point where the given code could be optimized. I chose to calculate the eigenvalues from the Jacobian directly so that future development of the code that will include more physical fields, such as those described in Chap. 6, would require fewer code modifications. When using the flux calculations to update the advanced time variable I needed to be cognisant of coordinate problems that arise when dealing with values as they approach the origin, r = 0, and axes, θ = 0, π/2. Although the origin is not a problem from the perspective of my system, since the event horizon provided a physical boundary that allowed me to ignore the origin, I developed the code to allow for the origin to be handled, which will be necessary if I study accretion onto a boson star, for example. To treat the problem with the radial coordinate at the origin we modified our differential operators [28], 1 ∂ ∂ → 3. r2 ∂r ∂r (B.1) Using this derivative operator I was no longer in danger of facing “divide-by-zero” errors near the origin. The problems with the axes were handled in a similar fashion, ∂ 1 ∂ →− , sin θ ∂θ ∂ cos θ (B.2) Again, we know sin θ = 0 on the axes, but cos θ = 1 on the axes, and thus the differential operator is not singular. Higher Dimensions Once the flux calculations were performed for one dimension I repeat the reconstruction and flux calculations for each spatial dimension investigated. While I stopped at two spatial dimensions for numerical efficiency, the methods developed are easily extended to three spatial dimensions. 256 B.3. MAIN ROUTINE Source With the flux contributions calculated I was left with calculating the source contributions to each cell. For the special relativistic calculations in Cartesian coordinates there are no sources. For the general relativistic calculations in curvilinear coordinates we needed to calculate the contributions to each cell due to the curved spacetime background. This was calculated using the techniques described in Chap. 3. The sources were calculated using both conservative and primitive variables. Primitive Variables The routines used to calculate the flux updates and source contributions performed the time evolution of the conservative variables. The next time step requires the primitive variables at the advanced time. To obtain these values I used the iterative single variable route finding method described in Sec. 3.3. There are several other techniques available as surveyed by Noble et al. [125], but I found the method selected to be best for the problem studied. The findings of [125] were that the 2-variable solver was optimal for more general systems. The use of root finders to determine the primitive variables given a set of conservative variables can lead to numerical errors. One such error includes finding unphysical roots which return a negative pressure or rest mass density. If left unchecked, these errors would destroy the simulations; consequently, I introduce an additional step before returning the calculated primitive variables. I use a method known as a “Floor” that checks the sign of the pressure and density calculated by the root finder. If either value is negative the Floor replaces the negative pressures and/or densities with a nominal small value. Once the pressure and/or density are altered I recalculate the conservative variables to ensure the system is consistent. I only needed the floor routine when simulating extreme systems where the flow had a high Mach number, and especially when the high Mach flows were past a near extremal (a ∼ 1) black hole. We described the use of a floor in more detail in Sec. 3.9.1. B.3.4 Update Boundary The boundary update routine was designed to update the ghost cells discussed in Sec. 3.9, whenever there was a physical boundary encountered. I implemented all the boundary conditions described in the thesis. All of these boundaries were necessary of the study presented. 257 B.4. FINAL REMARKS B.4 Final Remarks The above routines were compiled using the Intel compiler suite32 , as well as the gnu compilers. Both of these are freely available for non-commercial use. I found it advantageous to use different compilers for compilation. They allowed me to find potential coding errors that lead to run-time failures. Using the outline above one should be able to produce their own robust magnetohydrodynamic code. Since I use the hyperbolic divergence cleaning method that there is no particular special treatment of the magnetic field constraint outlined above. The introduction of a new conservative variable, while computationally more expensive, is logically cleaner, and leaves a lot less room for error when interpreting results. 32 http://software.intel.com/en-us/articles/intel-compilers/ 258
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Numerical analysis of the ultrarelativistic and magnetized Bondi--Hoyle problem Penner, Andrew Jason 2011
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Title | Numerical analysis of the ultrarelativistic and magnetized Bondi--Hoyle problem |
Creator |
Penner, Andrew Jason |
Publisher | University of British Columbia |
Date Issued | 2011 |
Description | In this thesis, we present numerical studies of models for the accretion of fluids and magnetofluids onto rotating black holes. Specifically, we study three main scenarios, two of which treat accretion of an unmagnetized perfect fluid characterized by an internal energy sufficiently large that the rest-mass energy of the fluid can be ignored. We call this the ultrarelativistic limit, and use it to investigate accretion flows which are either axisymmetric or restricted to a thin disk. For the third scenario, we adopt the equations of ideal magnetohydrodyamics and consider axisymmetric solutions. In all cases, the black hole is assumed to be moving with fixed velocity through a fluid which has constant pressure and density at large distances. Because all of the simulated flows are highly nonlinear and supersonic, we use modern computational techniques capable of accurately dealing with extreme solution features such as shocks. In the axisymmetric ultrarelativistic case, we show that the accretion is described by steady-state solutions characterized by well-defined accretion rates which we compute, and are in reasonable agreement with previously reported results by Font and collaborators [1,2,3]. However, in contrast to this earlier work with moderate energy densities, where the computed solutions always had tail shocks, we find parameter settings for which the time-independent solutions contain bow shocks. For the ultrarelativistic thin-disk models, we find steady-state configurations with specific accretion rates and observe that the flows simultaneously develop both a tail shock and a bow shock. For the case of axisymmetric accretion using a magnetohydrodynamic perfect fluid, we align the magnetic field with the axis of symmetry. Preliminary results suggest that the resulting flows remain time-dependent at late times, although we cannot conclusively rule out the existence of steady-state solutions. Moreover, the flow morphology is different in the magnetic case: additional features are apparent that include an evacuated region near the symmetry axis and close to the black hole. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-08-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-ShareAlike 3.0 Unported |
DOI | 10.14288/1.0072054 |
URI | http://hdl.handle.net/2429/36734 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2011-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-sa/3.0/ |
Aggregated Source Repository | DSpace |
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