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Numerical studies in gravitational collapse Akbarian Kaljahi, Arman 2015

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Numerical Studies in GravitationalCollapsebyArman Akbarian KaljahiB.Sc., Sharif University of Technology, 2008M.Sc., The University of British Columbia, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2015c© Arman Akbarian Kaljahi 2015AbstractIn the first part of this thesis, we solve the coupled Einstein-Vlasov system in spher-ical symmetry using direct numerical integration of the Vlasov equation in phasespace. Focusing on the case of massless particles we study critical phenomena inthe model, finding strong evidence for generic type I behaviour at the black holethreshold that parallels what has previously been observed in the massive sector.For differing families of initial data we find distinct critical solutions, so there is nouniversality of the critical configuration itself. However we find indications of atleast a weak universality in the lifetime scaling exponent, which is yet to be under-stood. Additionally, we clarify the role that angular momentum plays in the criticalbehaviour in the massless case.The second part focuses on type II critical collapse. Using the critical collapseof a massless scalar field in spherical symmetry as a test case, we study a general-ization of the BSSN formulation due to Brown that is suited for use with curvilinearcoordinates. We adopt standard dynamical gauge choices, including 1+log slicingand a shift that is either zero or evolved by a Gamma-driver condition. With bothchoices of shift we are able to evolve sufficiently close to the black hole threshold to1) unambiguously identify the discrete self-similarity of the critical solution, 2) de-termine an echoing exponent consistent with previous calculations, and 3) measurea mass scaling exponent, also in accord with prior computations. Our results canbe viewed as an encouraging first step towards the use of hyperbolic formulations inmore generic type II scenarios, including the as yet unresolved problem of criticalcollapse of axisymmetric gravitational waves.iiAbstractIn the last part, we present simulations of nonlinear evolutions of pure gravitywaves. We describe a new G-BSSN code in axial symmetry that is capable of evolv-ing a pure vacuum content in a strong gravity regime for both Teukolsky and Brillinitial data. We provide strong evidence for the accuracy of the numerical solver.Our results suggest that the G-BSSN is promising for type II critical phenomenastudies.iiiPrefaceAll of the work presented in this thesis, except the introduction chapter, are originalwork done by the Author and the research supervisor Matthew Choptuik. Chapter2 and 3 of this thesis are identical to their published versions [1, 2] (Phys. Rev. D90,104023, (2014) and Phys. Rev. D92, 084037, (2015)) with only minor changes to thetypesetting to fit the thesis format. Some footnotes are added for further explanationof some of the concepts.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Einstein’s Gravitational Field Equations . . . . . . . . . . . . . . . 41.3 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Gravitational Collapse: Black Hole Solution . . . . . . . . . . . . . 71.5 Critical Phenomena in Gravitational Collapse . . . . . . . . . . . . 101.6 3+1 Formulations of Einstein’s Equations . . . . . . . . . . . . . . . 141.6.1 ADM Decomposition . . . . . . . . . . . . . . . . . . . . . . 151.6.2 Recasting of ADM Equations: BSSN Formulation . . . . . . 211.7 Coordinate Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.8 Overview of Numerical Techniques for Time Dependent Problems . 311.9 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 33vTable of Contents2 Critical Collapse in the Spherically Symmetric Einstein - VlasovModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.1 Coordinate Choice and Equations for Metric Components . . 412.2.2 The Energy Momentum Tensor . . . . . . . . . . . . . . . . 422.2.3 Evolution of the Distribution Function . . . . . . . . . . . . 442.3 Static Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Evolution Scheme . . . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.4.3 Diagnostic Quantities and Numerical Tests . . . . . . . . . 522.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.5.1 Generic Massless Case . . . . . . . . . . . . . . . . . . . . . 572.5.2 Near-static Massless Case . . . . . . . . . . . . . . . . . . . . 682.5.3 Generic Massive Case . . . . . . . . . . . . . . . . . . . . . 762.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 783 Black Hole Critical Behaviour with the Generalized BSSN Formu-lation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.2.1 Generalized BSSN . . . . . . . . . . . . . . . . . . . . . . . 893.2.2 G-BSSN in Spherical Symmetry and Gauge Choices . . . . 953.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.3.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 993.3.3 Evolution Scheme and Regularity . . . . . . . . . . . . . . . 1013.3.4 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102viTable of Contents3.3.5 Finding Black Hole Threshold Solutions . . . . . . . . . . . 1073.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.4.1 Zero Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.4.2 Gamma-driver Shift . . . . . . . . . . . . . . . . . . . . . . . 1183.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.6 BSSN in Spherical Symmetry . . . . . . . . . . . . . . . . . . . . . 1223.7 Scalar Field Synamics and Energy-Momentum Tensor in SphericalSymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254 Non-linear Gravity Wave Evolutions with the G-BSSN Formula-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 Equations of Motion for Strong Gravity Waves Dynamics . . . . . . 1314.2.1 G-BSSN in Cylindrical Coordinate with Axial Symmetry . . 1344.2.2 Coordinate Choices . . . . . . . . . . . . . . . . . . . . . . . 1354.2.3 Note on Complexity and Regularity of the Equations . . . . 1364.2.4 Axisymmetric Initial Data . . . . . . . . . . . . . . . . . . . 1394.2.5 Brill Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . 1404.2.6 Teukolsky-type Initial Data . . . . . . . . . . . . . . . . . . . 1404.2.7 Computing the ADM Mass of the Gravitational Pulse . . . 1424.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.3.1 Numerical Grid . . . . . . . . . . . . . . . . . . . . . . . . . 1434.3.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 1494.3.4 Evolution Scheme . . . . . . . . . . . . . . . . . . . . . . . . 1514.3.5 Note on G-BSSN’s Additional Constraints . . . . . . . . . . 1524.3.6 Tests: Convergence of Primary Variables . . . . . . . . . . . 1534.3.7 Tests: Conservation of Constraints During Evolution . . . . 1554.3.8 Tests: Direct Validation via Einstein’s Equations . . . . . . 157viiTable of Contents4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.4.1 Evolution of Teukolsky-type Initial Data . . . . . . . . . . . 1594.4.2 Evolution of Brill Initial Data . . . . . . . . . . . . . . . . . 1604.5 Further Remarks and Conclusion . . . . . . . . . . . . . . . . . . . 1695 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A Appendix: FD, Finite Difference Toolkit . . . . . . . . . . . . . . 184A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184A.2 Overview of Finite Difference Method . . . . . . . . . . . . . . . . 187A.2.1 Computing the FDA Expression . . . . . . . . . . . . . . . . 190A.2.2 Iterative Schemes for Non-Linear PDEs . . . . . . . . . . . 193A.2.3 Testing Facilities: Convergence and IRE . . . . . . . . . . . 200A.3 Semantics of FD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209A.3.1 Parsing a PDE: Fundamental Data Type . . . . . . . . . . . 209A.3.2 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 211A.3.3 Initializing FD, Make FD, Clean FD . . . . . . . . . . . . . . 212A.3.4 Grid Functions Set: grid functions . . . . . . . . . . . . . 212A.3.5 Known Functions . . . . . . . . . . . . . . . . . . . . . . . . 214A.3.6 Valid Continuous Expression, VCE . . . . . . . . . . . . . . 214A.3.7 Valid Discrete Expression, VDE . . . . . . . . . . . . . . . . 215A.3.8 Conversion Between VDE and VCE . . . . . . . . . . . . . . 216A.4 Discretizing a PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 217A.4.1 Performing the Finite Differencing, Gen Sten . . . . . . . . 217A.4.2 Discretization Scheme, FD table . . . . . . . . . . . . . . . 218viiiTable of ContentsA.4.3 Changing the FDA Scheme: FDS, Update FD Table . . . . . 219A.4.4 Accessing the FD Results: Show FD . . . . . . . . . . . . . . 222A.4.5 Defining Manual Finite Difference Operators: FD . . . . . . . 224A.5 Posing a PDE & Boundary Conditions Over a Discrete Domain . . 225A.5.1 Discrete Domain Specifier: DDS . . . . . . . . . . . . . . . . . 226A.5.2 Imposing Outer Boundary Conditions . . . . . . . . . . . . . 228A.5.3 Periodic Boundary Condition: FD Periodic . . . . . . . . . 230A.5.4 Implementing Ghost Cells for Odd and Even Functions: A FD Odd,A FD Even . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231A.6 Solving a PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236A.6.1 Creating Initializer Routines: Gen Eval Code . . . . . . . . . 236A.6.2 Point-wise Evaluator Routines with DDS: A Gen Eval Code . 238A.6.3 Creating IRE Testing Routines: Gen Res Code . . . . . . . . 240A.6.4 Creating Piece-wise Residual Evaluator Routines . . . . . . . 240A.6.5 Creating Solver Routine: A Gen Solve Code . . . . . . . . . 241A.6.6 Communicating with Parallel Computing Infrastructure . . . 242A.6.7 Example: Crank-Nicolson Implementation of Wave Equation 243A.7 List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 245ixList of Tables2.1 Families of generic initial data . . . . . . . . . . . . . . . . . . . . . . 562.2 Summary of measured lifetime scaling exponents for the massless . . 672.3 Measured lifetime scaling exponent . . . . . . . . . . . . . . . . . . . 732.4 Summary of measured lifetime scaling exponents for the massive . . 78xList of Figures1.1 Coordinate system in the 3+1 decomposition . . . . . . . . . . . . . 162.1 A portion of the discretized computation domain . . . . . . . . . . . 482.2 Results of various diagnostic tests . . . . . . . . . . . . . . . . . . . . 542.3 Snapshots of the distribution function . . . . . . . . . . . . . . . . . 582.4 Snapshots of the distribution function for a near-critical calculation . 592.5 Time evolution of ‖∂ta(t, r)‖2 . . . . . . . . . . . . . . . . . . . . . . 622.6 Lifetime scaling of near-critical configurations . . . . . . . . . . . . . 632.7 Radial metric function a(r˜) at criticality . . . . . . . . . . . . . . . . 642.8 Lapse function α(r˜) at criticality . . . . . . . . . . . . . . . . . . . . 652.9 Sample static phase space configurations . . . . . . . . . . . . . . . . 702.10 Plots of the radial metric function . . . . . . . . . . . . . . . . . . . 712.11 The value of Γ = maxr(2m/r) versus central redshift . . . . . . . . . 722.12 Lifetime scaling computed from families of initial data . . . . . . . . 753.1 Results from various tests . . . . . . . . . . . . . . . . . . . . . . . . 1033.2 Echoing behaviour in the scalar field . . . . . . . . . . . . . . . . . . 1103.3 The maximum central value, Rmax, of the four-dimensional Ricci scalar1113.4 Discrete self-similarity of the geometry of spacetime . . . . . . . . . 1133.5 Snapshots of radial mass density . . . . . . . . . . . . . . . . . . . . 1143.6 Profiles of matter and geometry variables . . . . . . . . . . . . . . . 1173.7 Profiles of various G-BSSN variables . . . . . . . . . . . . . . . . . . 119xiList of Figures4.1 Distribution of grid points on a non-uniform grid . . . . . . . . . . . 1454.2 Initial profile of the conformal metric component . . . . . . . . . . . 1474.3 Convergence factor for the G-BSSN variables . . . . . . . . . . . . . 1544.4 Evolution of the conserved variables . . . . . . . . . . . . . . . . . . 1564.5 Convergence of the Einstein equations residuals . . . . . . . . . . . . 1584.6 Evolution of a non-linear Teukolsky-type wave packet . . . . . . . . 1614.7 Collapse of Teukolsky-type wave initial data . . . . . . . . . . . . . . 1624.8 Non-linear evolution of a Teukolsky-type wave packet . . . . . . . . 1634.9 Typical evolution of collapsing Teukolsky-type data . . . . . . . . . . 1644.10 Time evolution of central lapse . . . . . . . . . . . . . . . . . . . . . 1654.11 Dispersal evolution of Brill initial data . . . . . . . . . . . . . . . . . 1664.12 Collapse of Brill data . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.13 Central lapse for Brill data evolution . . . . . . . . . . . . . . . . . . 168A.1 Five points specifying the FDA scheme . . . . . . . . . . . . . . . . . 219A.2 Specifying different types of FD schemes . . . . . . . . . . . . . . . . 220xiiAcknowledgmentsI would like to thank my research supervisor Matthew Choptuik for his guidance andtremendous support during the research work of my PhD program. I would also liketo express my gratitude to other members of my PhD committee: William Unruh,Jeremy Heyl and Colin Gay for their insightful comments and thorough reading ofa draft of this thesis. My colleagues and friends in numerical relativity group inUBC, Silvestre Aguilar, Graham Reid and Daoyan Wang have provided me withmany helpful discussions and created a fun and stimulating work environment, andI am very thankful for that. Many thanks to my lovely friends, Joy Peng and MiyaGu, who have been my little family in Canada and brought lots of laughters andcats to my life during my PhD program. Lastly, I cannot thank my family enoughfor their support and encouragement. The majority of the simulations performed toobtain the results in this thesis are done on Westgrid cluster. Finally, I would liketo acknowledge the financial support from UBC via FYF scholarship for my PhDprogram.xiiiChapter 1IntroductionEinstein’s theory of gravity relates the geometry of the spacetime to its matter con-tent. The spacetime in General Relativity is modeled as a 4-dimensional Lorentzianmanifold. On this manifold, gravity is identified as a characteristic of its 4 dimen-sional geometry (curvature), whereas in Newton’s theory, gravity is a field definedon a 3 dimensional flat space and time is an independent coordinate (universal to allobservers). In this geometric interpretation, the “gravitational interaction” of parti-cles is solely by experiencing the curved spacetime as they move along the geodesicsof the curved spacetime.As one might expect, in the limit where the deviation of the spacetime metricfrom a flat metric is large, the general relativistic geometric equations can no longerbe approximated with Newton’s field equation. This limit is related to the con-centration of matter, or equivalently one can define a characteristic “gravitationallength”, LG, that depends on the total mass of the system1 as:LG =Gc2M . (1.1)Then the typical size of the system can be compared to LG, and for a system withL ≈ LG (a highly compact object relative to a typical star) the non-linear effectsin General Relativity become prominent and the structure and dynamics of thespacetime is referred as a strong gravity scenario.2 In this regime, one of the most1This characteristic length for the Sun is 1.47 km.2Of course, another parameter that distinguishes “classical” Newtonian system from a relativisticsystem is its particles typical speed compared to the speed of light: v/c.1Chapter 1. Introductionremarkable predictions of General Relativity is the black hole solution—a region ofspacetime that cannot causally affect the outside world.Despite the early discovery (1916) of the black hole solution, only few otherphysically relevant analytic solutions have been found, since the proposal of theGeneral Relativity theory by Einstein (1915). This is mainly due to the highly non-linear and complex nature of the geometric equations that describe the gravitationalinteraction. This complexity has spurred the development of numerical relativity,where large scale computing is used to model strong gravity scenarios. The mainconcentration of studies in numerical relativity have been on two aspects of stronggravity: 1) the astrophysically relevant scenarios, particularly in the context ofcompact objects dynamics and mergers, and 2) fundamental studies in the theory ofGeneral Relativity. Most of the research in this thesis is in the later spirit. We referthe reader to [3] for a recent and extensive review of the frontier of the numericalrelativity field.The fundamental studies of General Relativity in a “numerical lab” were pio-neered by Choptuik [4] who numerically discovered an unexpected emerging phe-nomena at the threshold of black hole formation in the collapse of a massless scalarfield. Following works in the collapse of various matter sources unveiled similarrich phenomenology, as the gravitational strength of the matter source is tuned toevolve precisely at the boundary of two possible classes of final states: black holeformation or dispersal (flat space). With features such as the apparent universalityof the solution and power-law scaling, an analogy to phase transitions in statisticalphysics was established, thus the name: critical phenomena in gravitational collapse.The research work of this thesis is mainly focused on critical phenomena and is acontribution to fundamental studies in gravitational physics. References [5] and [6]provide extended reviews of studies in critical phenomena in gravitational collapse.and a more recent overview of the field can be found in [3]. A formal description ofthe threshold solution and its properties will follow shortly, and all relevant critical2Chapter 1. Introductionphenomena studies to the projects in this thesis will be reviewed in the introductionsection of each chapter.Besides the existence of black hole solutions, another prediction of GeneralRelativity—that does not have a counterpart in Newtonian gravity—is the exis-tence of gravitational waves. Gravitational waves may be viewed as ripples in thegeometry of spacetime that propagate at a finite speed, namely that of light. Directphysical detection of gravitational waves is an ongoing global effort in physics. [7–11]The expected amplitude of a gravitational wave pulse that would reach the Earthis extremely low and can be studied in the linearized approximation. However, thesource of the wave is believed to be in the regime of strong gravity dynamics, suchas the merger of two black holes. The pioneering works in numerical relativity inthis context are the very first successful long term evolution of binary black holemergers by Pretorius (2005) [12] followed by Campanelli et.al. (2006) [13] and theextraction of the gravitational wave form by Baker et.al. (2006) [14]. For a reviewof the state of the art research in this topic see: [3].Gravity waves are another topic of interest in this thesis, where again we adopta fundamental perspective and focus on the numerical studies of the strong gravityregime where a gravitational wave packet can collapse and form a black hole. Thispart of the thesis is inspired by some unresolved questions in critical collapse of puregravity waves, as we will discuss extensively in the next chapters.We note that Chapter 2 and 3 of this thesis are published studies and there-fore are written in a self-contained manner (to the extent that the standards ofpublication allow). Chapter 4 is also an independent project, but the theoreticalformulation of it heavily depends on Chapter 3. Each chapter contains an introduc-tory section where we introduce the research problem and review the literature onthat topic. The rest of the current chapter aims to provide a general introductionto the formulations and techniques used in the rest of the thesis. In particular, wecontinue with: a formal presentation of Einstein’s field equations, a description of31.1. Notationgravity waves and black hole solutions, an overview of critical phenomena, a sum-mary of two formulations of Einstein’s equation used in numerical relativity as wellas the choice of coordinates, and a quick overview of the numerical methods used inthe thesis (which are more extensively discussed in the Appendix).1.1 NotationIn this thesis we adopt units where Newton’s constant G and the speed of light care set to 1. The spacetime metric is chosen with the signature −+++ and all thesign conventions are similar to Misner et al.[15]. The Latin alphabets {a, b, c, · · · , h}are used for abstract indexing, introduced by Wald [16], of both 4-dimensional andspatial 3-dimensional tensors. We use Latin indices starting from i: {i, j, k, · · · , n}(that runs from 1 to 3) to specifically denote the purely spatial 3-dimensional tensors.The Greek indices (that run from 0 to 3) are used to denote the components of the4-dimensional tensors in a specific coordinate choice. The Einstein’s conventionis adopted throughout the thesis, whereby the summation over repeated indices isassumed. ∇ denotes the 4-dimensional covariant derivative associated with the 4-metric, while D denotes the 3-dimensional covariant derivative associated with theinduced 3-metric on the spatial hypersurfaces in a 3+1 decomposition. Parenthesesenclosing the indices of a tensor denote symmetrization of the tensor, for exampleA(ij) = (Aij +Aji)/2.1.2 Einstein’s Gravitational Field EquationsThe geometric description of the gravitational field is encoded in the 4-dimensionalmetric tensor, gab, where one dimension , identified as time, has the opposite signto that of the rest of the 3 spatial dimensions. This metric measures the Lorentzian41.2. Einstein’s Gravitational Field Equationsdistance between two spacetime points:ds2 = gabdxadxb , (1.2)where xa labels the coordinates and dxa is the difference vector between two nearbypoints. A special case of the metric is the flat metric (Minkowski spacetime) and inCartesian coordinate where x0 = t and xi = (x, y, z) is given by:ηab = diag(−1, 1, 1, 1) . (1.3)The structure of the spacetime (given by the metric gab) is governed by Einstein’sequation:Gab = 8πTab , (1.4)where Gab is the Einstein tensor and schematically depends on the metric gab andits first and second space and time derivatives. Tab is the energy-momentum tensorand constitutes all of the non-gravitational energy and momentum contributions ofmatter sources that are present in the spacetime.The Einstein tensor is given in terms of the 4-Ricci tensor, Rab and its trace,R = Raa, as:Gab = Rab − 12gabR . (1.5)The Ricci tensor is constructed from the Riemann tensor by the contraction:Rab = Rcacb . (1.6)The Riemann tensor measures the curvature of the spacetime and is defined by itsaction on a covariant vector as:Rdabcvd = ∇b∇cva −∇c∇bva . (1.7)51.3. Gravitational WavesIt vanishes identically if and only if the spacetime is flat. The explicit form of theRiemann tensor in terms of the metric gab can be written using the definition of theChristoffel symbols, Γabc,Γabc =12gad(∂cgdb + ∂bgdc − ∂dgbc) . (1.8)Specifically, the Riemann tensor components are:Rabcd = ∂cΓabd − ∂dΓabc + ΓaecΓebd − ΓaedΓebc . (1.9)As one can see from (1.8,1.9) and the definition of Einstein tensor, the Einstein’sequation becomes a set of 10 second order PDEs for the metric components gab.The properties of these equations will become clearer as we introduce the linearizedapproximation and the 3+1 decomposition of the equations. We note that thereare several in-depth introductory textbooks in General Relativity such as Misner etal. [15], Weinberg [17], Wald [16] and Carroll [18]. The reader may refer to thesetextbooks for further explanations of the concepts introduced in this section.1.3 Gravitational WavesAs mentioned before, one aspect of General Relativity that is absent from Newtoniangravity is the finite-speed propagation of disturbances in the gravitational field. Thespacetime ripples—small changes in the metric gab—will carry information abouttheir source and in a region far away from their origin, they can be modeled asperturbations of the Minkowski spacetime. Consider a small deviation from the flatmetric in Cartesian coordinate:gab = ηab + hab , (hab ≪ 1) , (1.10)61.4. Gravitational Collapse: Black Hole Solutionand define the trace-reversed perturbation metric, h¯ab, as:h¯ab = hab − 12ηabh , (1.11)where h ≡ ηcdhcd is the trace of the perturbed metric. Using the linearized coordi-nate freedom, one can further impose the condition:∂ah¯ab = 0 . (1.12)In this gauge choice, the linearized Einstein’s equation in vacuum (Tab = 0) is thengiven by:G(1)ab = ∂c∂ch¯ab = 0 (1.13)where superscript (1) denotes the approximation where we only keep the terms in-volving h¯ab and its derivatives up to first order. This is a wave equation for thecomponents of the perturbation metric, h¯ab, and illustrates the wave-like character-istics of the Einstein’s equation. More discussions on linearized gravity waves canbe found in standard texts on General Relativity such as [15, 16].1.4 Gravitational Collapse: Black Hole SolutionA black hole is a region of spacetime that cannot signal information to the outsideworld. This region is a remnant of a gravitational collapse process where the strengthof the self-gravitation of the matter is increasing as the matter is compressed to arapidly shrinking region of the spacetime. Eventually, if the length scale of the sys-tem can reach to the order of its gravitational length, L ≈ LG, (as introduced in(1.1)) the system collapses to a black hole. From the geometrical point of view, allof the causal curves – including the null geodesics, the path of photons – becomeconfined in the black hole region. The boundary of this region that causally discon-nects the interior from the outside world is called the event horizon. It is known71.4. Gravitational Collapse: Black Hole Solutionthat black holes are indeed present in our universe: the super-massive black holesin the center of most galaxies [19, 20], and black holes as the final fate of massivestars [21].One example of a spherically symmetric spacetime that contains a black hole isgiven by the Schwarzschild metric:ds2 = −(1− 2Mr)dt2 +(1− 2Mr)−1dr2 + r2dθ2 + r2 sin2 θdϕ2 . (1.14)Here the event horizon is located at rs ≡ 2M and rs is known as the Schwarzschildradius. 3 The interior of the black hole, r < rs, cannot physically affect the outsideworld, r > rs. The solution (1.14) is known as an eternal black hole, since it existsfor all time −∞ < t <∞.This solution also demonstrates the coordinate dependence of the metric com-ponents, where here grr is divergent at the event horizon r = rs = 2M . However,this pathology is purely due to the choice of coordinates and can be removed by acoordinate transformation. For example, the Schwarzschild solution can be writtenin an isotropic radial coordinate, 4 where it takes the form:ds2 = −(1−M/2r˜1 +M/2r˜)2dt2 +(1 +M2r˜)4(dr˜2 + r˜2dθ2 + r˜2 sin2 θdϕ2) . (1.15)In this coordinate system, the horizon is located at r˜s = M/2, and there is nodivergence of the metric component grr at r˜ = r˜s.5In addition to the existence of the horizon, another important property of theblack hole solution is that it contains a spacetime singularity. This singularity is atrue geometric pathology of the spacetime and is manifested as divergence of certain3For a system with a mass equal to the Sun, the Schwarzschild radius is twice its gravitationallength introduced in (1.1), i.e 2.94 km4The coordinate transformation between the isotropic radial coordinate and the radial coordinateassociated with (1.14)—known as the areal coordinate—is given by: r = r˜(1 +M/2r˜)2.5Note that the gtt is zero at the horizon, which is a coordinate pathology, however this form ofthe metric can be used for numerical simulations.81.4. Gravitational Collapse: Black Hole Solution(coordinate independent) geometric invariants. 6 For example, for the Schwarzschildmetric (1.14) the Kretschmann scalar is given by:I ≡ RabcdRabcd = 48M2r6(1.16)and clearly diverges at r → 0. Such singularities can also form from a gravitationalcollapse process, but whether they can form outside of event horizons in astro-physical scenarios is yet an open problem. The conjecture that an event horizonalways forms to hide the singularity from the outside world is known as the cosmiccensorship hypothesis [22].The confinement of the singularity by an event horizon has a practical implicationfor numerical simulations of black hole dynamics or gravitational collapse. Of course,no numerical solver can perform evolutions on a domain that contains a singularitywhere the dynamical variables diverge. However, the event horizon isolates theobservable numerical domain (outside of the black hole) from the singularity bydisconnecting the physical characteristics of the equations inside from the outside ofthe black hole. In another word, no physical effect can be transferred to the outsidesince no causal curve can emerge from within the horizon. This effectively allowsan arbitrary choice of the geometry inside of the black hole. 7 Therefore, assumingthat no singularity forms outside of an event horizon, a numerical evolution of acollapse scenario can be carried out from a spacetime with no black hole to a onethat contains a black hole which eventually settles to a stationary solution. In effect,cosmic censorship allows us to obtain the spacetime solution for the exterior of theblack hole even if we do not have a theory to describe the physics of the singularity.6Another example of a geometric singularity is a point of geodesic incompleteness where thegeodesics of spacetime “end” at a finite affine parameter [16]. Spacetime singularity is been provento always exist inside of a black hole [16].7Under the assumption that the numerical method respects causality, for example the system isevolved via a set of hyperbolic equations.91.5. Critical Phenomena in Gravitational Collapse1.5 Critical Phenomena in Gravitational CollapseCritical phenomena arise in various systems of matter coupled to Einstein gravity(or pure gravity waves) through the dynamical construction of spacetime solutions.The construction starts with initialization of the matter content of the spacetime,some of whose properties are controlled by a parameter p. This parameter controls,for example, the self-gravitational strength of the system, and can be chosen arbi-trarily. For instance, p can be the amplitude of a pulse of matter field, or the averagevelocity of a collection of particles. Naturally, this parameter also labels each of thedynamical spacetime solutions that are constructed by numerically evolving the sys-tem. A parameter survey over this family of spacetime solutions is then performedby numerically evolving initial configurations defined by various values of p. Thevalue of p is assumed to vary in an interval with two end points: 1)“weak”: corre-sponding to a sufficiently small value of p that the system’s self-gravitation remainsweak. The matter typically disperses to infinity leaving flat spacetime behind8;2) “strong”: corresponding to a sufficiently large value of p that gravity is strongenough to cause the system to collapse to a black hole. As one might speculate,these two regimes have a point of transition along p, namely the critical value p⋆. 9For p > p⋆ the final state is a black hole with mass MBH(p), while p < p⋆ does notresult in black hole formation, i.e. MBH = 0. The solution associated with p = p⋆is referred as the black hole threshold solution (or critical solution). Operationally,the value p⋆ can be found numerically using a binary search10 and its accuracy islimited by the numerical resolution, and often (for sufficiently high resolution) canbe pushed close to the machine precision, ≈ 10−16. The characteristics of the space-time and matter configuration at and near the precise threshold value, p⋆, comprises8Another scenario is that a bound, stable solution such as a star is formed.9One can imagine another possibility where the corresponding values of p for weak and stronginitial data spread over the interval with no defined single boundary. Such scenario will result ina “chaotic” behaviour, where the slightest change in the tuning parameter causes the final stateto deviate drastically. The fact that the two regimes are disjoint by a single value is somewhat anon-trivial observation and suggests that gravitational collapse in General Relativity is not chaotic.10We will formally describe this process in Chapter 2 and 3101.5. Critical Phenomena in Gravitational Collapsethe main subject of black hole critical phenomena.In the numerical lab, the characteristics of the critical solution emerge only to acertain extent, i.e. as the tuning process (binary search) is performed to drive p→ p⋆more closely, the features of the underlying critical solution can be observed onlypartially.11 As will become clear shortly, fully capturing these features numerically(for a certain type of critical solution) requires infinite resolution and imposes severechallenges for a numerical solver. The extreme requirement for the resolution isdue to the existence of fine structure in the critical solution. Historically, the firstsuccess in numerically exploring the critical regime by Choptuik [4] relied on theimplementation of an Adaptive Mesh Refinement (AMR) algorithm that providedthe needed resolution.Black hole critical phenomena can be divided into two broad categories by con-sidering the mass of the final black hole,MBH, as a function of the tuning parameter,p. In the weak or sub-critical regime, we simply have MBH(p < p⋆) = 0, while forp > p⋆, namely the super-critical regime, there are two possibilities: 1) the massof the black hole increase continuously from zero as p crosses the value p⋆; or 2)the mass function has a finite gap. In analogy with phase transitions in thermody-namical systems, MBH can be viewed as an order parameter and accordingly, thecontinuous transition is referred as Type II critical collapse while the critical collapsewith a mass gap is named Type I. All of the type II critical solutions discovered sofar, and some of the type I solutions, exhibit a common feature: universality. Thisuniversality is observed in the numerical tuning experiments, where there is uniquefinal configuration (of the spacetime geometry as well as the matter distribution)that is independent of the choice of parameter p and, more generally, the initialconfiguration of the matter. This observation signals the existence of a unique (orat least isolated in function space) critical solution in the function-space of all so-11For example, only a finite number of echoes (periodic behaviour in a logarithmic radial coor-dinate) can be observed rather than infinitly many of them that are present in the exact criticalsolution.111.5. Critical Phenomena in Gravitational Collapselutions. Moreover, the critical solution is an intermediate attractor point withinthe sub-function-space labeled by p = p⋆. The dynamical-system point of view ofthese critical phenomena is further discussed in [5, 23]. Beside universality, criticalsolutions tend to be 1-mode unstable and the unstable eigenvalue of the universalsolution, in both types, is related to a measurable exponent in the critical solutionas we discuss in the following.Type II Critical PhenomenaIn type II collapse, the mass of the black holes that form is often given by:MBH = |p− p⋆|γ , (1.17)The constant γ is known as the mass-scaling exponent and is universal for a givenmatter model, i.e. is independent of the details of the initial data. The massscaling (1.17) indicates that one can create a black hole of arbitrarily small size byapproaching p→ p⋆ from the super-critical regime.Type II critical solutions also exhibit scale-invariance or self-similarity. A self-similar solution can be written as a function Z⋆(τ, x) in which τ = − ln(t− t⋆)—thelogarithm of the proper time measured from the time, t⋆, that the critical solutionforms, usually referred as the accumulation time—x is the scale invariant coordinate,x = r/(t − t⋆), and Z denotes some function of the primary dynamical fields. Inthis notation, a continuous self-similar (CSS) solution is defined as:Z⋆(τ, x) = Z⋆(x) , (1.18)and a discrete self-similar (DSS) solution is:Z⋆(τ +∆, x) = Z⋆(τ, x) . (1.19)An example of discrete self similarity (DDS) arose in the original work on a121.5. Critical Phenomena in Gravitational Collapsemassless scalar field collapse by Choptuik. Another way to represent the discreteself similarity of the critical solution in time and spatial coordinates is:Z⋆(τ, r) = Z⋆(τ + n∆, en∆r) , (1.20)in which the notion of the echoing behaviour of the solution is clearer. After eachecho, τ → τ+∆, the solution repeats itself on a scale that is e∆ smaller. In addition,from (1.20), in every snapshot of the solution, t = const, there is a periodic behaviourin the profile of the matter field as a function of ρ = ln(r), with period ∆. A similarrepetitive structure forms in the geometry of spacetime as well. The constant ∆ isknown as the echoing exponent and, as for the scaling exponent, is universal for aspecific matter source. This echoing behaviour results in the formation of structurein the solution on ever smaller scales, and as we mentioned in the previous section,requires fine numerical resolution, i.e. adaptive distribution of grid points toward thecentral collapse region. Finally, we note that as the self-similar solution approachesthe accumulation point t → t⋆, or τ → ∞, in the continuum limit , i.e. for aprecisely critical solution p = p⋆, the curvature diverges at the radial accumulationpoint (center of the collapse in symmetric cases) and a naked singularity forms.Chapter 3 and 4 of this thesis are focused in type II critical collapse studieswhere we explore the applicability of a popular formulation of Einstein’s equationin the context of type II DSS critical collapse scenarios. There, we will revisit typeII critical phenomena and discuss the features of the massless scalar field thresholdsolution.Type I Critical PhenomenaIn critical collapse with a mass gap,MBH(p→ p⋆+) > 0, the solution approachesan intermediate state (between collapse and dispersal) that is static (or periodic)and unstable. An initial configuration that is tuned to p ≈ p⋆ exhibits a time-scalingbehaviour: the dynamically evolving solution approaches the critical static solutionand spends an increasingly large amount of time in the vicinity of it. This time131.6. 3+1 Formulations of Einstein’s Equationsscaling is given by:τ = −σ ln |p − p⋆| (1.21)in which σ is known as time-scaling exponent, and similar to the exponents in TypeII critical collapse, can be universal for certain matter types. This scaling behavioursuggests that the critical solution possesses a time translation symmetry.In addition to the dynamical construction of type I/II critical solutions via atuning process, critical solutions can be constructed directly from an ansatz to thecoupled Einstein-matter system that reflects the symmetry of the solution (self-similar or static). In turn, the scaling laws (1.17,1.21) can be understood from theperspective of perturbation analysis [24–26]. The key observation here is that theexistence of the universal exponent (σ for type I and γ for type II) can be explainedby the existence of only one unstable mode in the critical solution with growth factor(Lyapanov exponent), λ. In fact, the exponent is directly related to the growingmode and is simply equal to its inverse, 1/λ.Chapter 2 of this thesis is concerned with type I critical collapse in the Einstein-Vlasov system, and answers some unresolved issues concerning this model. Extensivereviews of all the work in type I and II critical collapse can be found in [5, 23] andthe more recent reference [3].1.6 3+1 Formulations of Einstein’s EquationsIn General Relativity, the evolution of the spacetime geometry is given by a setof second order partial differential equations—as we observed in the approximatelinearized form (1.13). Specifically, Einstein’s equation can be posed as a standardinitial value problem (Cauchy problem): loosely speaking, the solution is expectedto be uniquely determined for given “position”, i.e gab, and “velocity”, i.e. ∂tgab atthe initial time. Schematically, the Einstein’s equation then determines the secondtime derivative of the metric, ∂2t gab, which can be integrated forward for the given141.6. 3+1 Formulations of Einstein’s Equationsinitial data. To pose the Einstein’s equation as a Cauchy problem, one needs afoliation of the spacetime to 3 dimensional space-like surfaces labeled by a timecoordinate t. Such process is usually referred as a 3+1 decomposition of Einstein’sequation. In this section we review the classic ADM [27] decomposition and alsointroduce a recasting of it, the BSSN formulation, that is particularly suitable fornumerical computations and that has become the most popular 3+1 formulation inthe numerical relativity community. Later in Chapter 3 and 4, a generalization ofthe BSSN formulation will be introduced and implemented to perform numericalsimulations.We note that the 3+1 casting of Einstein’s equations further reveals the internalconstraints in the field equations. They are not trivially apparent in the covariantform of the Einstein’s equations. These equations are not of evolutionary type,but rather constrain the initial data: gab and ∂tgab. Therefore the Cauchy problemcannot be posed for a arbitrary initial data; rather a set of elliptic-type constraintequations must be solved at the initial time.1.6.1 ADM DecompositionIn the 3+1 ADM decomposition of the 4 dimensional spacetime manifold (M,gµν),one assumes that there exist a family of disjoint 3 dimensional space-like hyper-surfaces, Σt, which can be considered as level surfaces of a scalar function t. Thevariable t can be interpreted as the global time function and using the gradient ofthe function t one can define the normal vector, na, to the hypersurface Σt, as shownin Fig. 1.1:na = −αgab∇bt . (1.22)We assume ∇at is non-zero everywhere and α is the normalization factor:||∇at||2 = gab∇at∇bt ≡ − 1α2, (1.23)151.6. 3+1 Formulations of Einstein’s Equations............................................... ................................................................ ................................................................................................................................................................................... ....................................................................................................... ...................................................................................................................... ................................................................ ................................................................................................................................................................................. ....................................................................................................... ......................................................................Σt+dtΣt✻✁✁✁✁✁✁✕✏✏✏✶αnadttadtβadt(t, xi)(t + dt, xi)................................ ...............................Figure 1.1: Coordinate system in the 3+1 decomposition of Einstein’s equation. nµdenotes the normal vector, while ta represents the unit coordinate vector along thetime coordinate t. These two vectors do not necessarily coincide and their deviationbetween two hypersurfaces Σt and Σt+dt defines the the shift vector, βa, which is byconstruction purely spatial.and is referred to as the lapse function. The space-like surfaces Σt, with a givenspatial coordinate system xi on them, constitute the 3+1 slicing of the 4-dimensionalmanifold M where each point in the spacetime is assigned coordinates (t, xi). Thetime coordinate vector, ta = ∂/∂t, arises from the global time function and can bewritten as a linear combination of the normal vector, na, and a purely spatial vector:βa:ta = αna + βa . (1.24)In this linear combination (also illustrated in Fig. 1.1), the coefficient of na is thelapse function, α. This can be easily seen by inserting the definition of na (1.22) inthe identity: ta∇at = dt/dt = 1, and using the fact that by definition we choose βato be a spatial (i.e. it lives on the hypersurface, Σt), therefore:naβa = 0 . (1.25)161.6. 3+1 Formulations of Einstein’s EquationsThe vector βa is referred as the shift vector and can be explicitly written as a 3dimensional vector:βa = (0, βi) , (1.26)in the coordinate system (t, xi).Using (1.24), one can compute the components of the normal vector, nµ, in thecoordinate system:nµ =(1α,− 1αβi). (1.27)Using the normal vector we can, in turn, can build the induced 3-metric on the thehypersurfaces, Σt:γab = gab + nanb . (1.28)The tensor γab is purely spatial, (as can be seen simply by verifying that γabna = 0)and it measures the distances within the spatial surface Σt. In addition, the innerproduct of any two purely spatial vectors, (vi, wj), computed using γab is identicalto the inner product computed with the 4 dimensional metric, gab; therefore γabis indeed the induced metric. The full spacetime metric, gab, can be reconstructedusing the 3-metric and the lapse and shift:gµν = −α2 + βlβl βiβj γij , (1.29)Using the normal vector na, one can define the projection tensor:γ ba = gba + nanb = δ ba + nanb (1.30)which projects 4-vectors to their spatial component on the hypersurface Σt. Theprojection of higher rank tensors into the spatial hypersurfaces can be achievedby contracting each index with the projection operator. This operator is usually171.6. 3+1 Formulations of Einstein’s Equationsdenoted by the symbol ⊥. For example,⊥ Aab ≡ γ ca γ db Acd (1.31)and ⊥ Aab is a purely spatial tensor.One particular projection that characterizes the embedding of the hypersurfacesΣt in the spacetime is the projection of the gradient of the normal vector: ∇cnd.Specifically, the extrinsic curvature of the surface Σt is defined as the projection ofthe negative gradient of its normal vector:Kab ≡ −γ ca γ db ∇cnd = −12Lnγab , (1.32)where L denotes the Lie derivative. Here the last equality is due to the identityLngab = ∇anb+∇bna and relation (1.28) between the 3 metric and 4 metric. Usingthe last identity in (1.32) and the linearity of the Lie derivative:∂t = Lt = Lαn+β = αLn +Lβ , (1.33)one can see that the definition of the extrinsic curvature can be written as:(∂t −Lβ)γab = −2αKab . (1.34)Therefore the extrinsic curvature can be seen as generalization of the “time deriva-tive” of the 3 metric γab. Using this definition, Einstein’s equations (1.4) can bereduced to a set of first order equations where the time evolution of the 3 metricis given by the extrinsic curvature (by definition) while the time evolution of theextrinsic curvature is given by the field equation projected to the spatial hypersur-faces12.The energy-momentum tensor on the left hand side of (1.4) can be projected12Combined with constraint equations.181.6. 3+1 Formulations of Einstein’s Equationsinto 3 parts:ρ = nanbTab , (1.35)Sa = −γabncTcb , (1.36)Sab = γacγbdTcd , (1.37)and accordingly, Einstein’s equation relates these quantities to the dynamical quan-tities of 3+1 decomposition: the 3-metric and the extrinsic curvature. The equa-tions can be found by relating the 4 dimensional Riemann tensor Rabcd to the 3dimensional Riemann tensor and the extrinsic curvature of the spatial slice Σt. Theprocess results in a set of identities known as Gauss-Codazzi equations. This ratherlengthy but straightforward calculation is now part of standard textbooks in numer-ical relativity and we refer the reader to [28] for details of the derivation. Here, weonly describe the results. The Einstein tensor contraction, nanbGab (1.35) results inscalar equation known as the Hamiltonian constraint,H = R+K2 −KijKij − 16πρ = 0 , (1.38)where R is the 3-Ricci scalar associated with the 3-metric γij and K is the trace ofthe extrinsic curvature:K ≡ γijKij . (1.39)The contraction γabncGcb (1.36) results in a spatial vector equation, known as themomentum constraint:Mi = Dj(Kij − γijK)− 8πSi = 0 , (1.40)in which Dj denotes the covariant derivative associated with the 3-metric, γij. The191.6. 3+1 Formulations of Einstein’s Equationsequations (1.38,1.40) are referred to as constraint equations since they do not containany second time derivative and only relate the “positions” and “velocities”, i.e. gijand Kij on each spatial slice Σt.Finally the contraction γacγbdGcd (1.37) provides the equation for the generalizedtime derivative of the extrinsic curvature, (∂t−Lβ)Kij . Combined with (1.34), theADM evolution equations for the 3-metric and the extrinsic curvature are:∂tγij = −2αKij +Diβj +Djβi , (1.41)∂tKij = −DiDjα + α (Rij − 8πSij + 4π(S − ρ)γij) + α(KKij − 2KikKkj)+ βk∂kKij +Kik∂jβk +Kkj∂iβk , (1.42)in which we have explicitly expanded the Lie derivative terms. Again, Di denotethe covariant derivative associated with the 3-metric γij and similarly Rij is the3-Ricci tensor associated with the 3-metric. This completes the process of castingEinstein’s equations as a 3+1 Cauchy problem, where the initial data {gij ,Kij}that satisfy the Hamiltonian and momentum constraints can be integrated forwardin time using (1.41,1.42) to find the geometry of spacetime. We note that a formal in-depth derivation of 3+1 formulations of Eintein’s equation can be found in numericalrelativity textbooks such as Baumgarte et al. [28], Gourgoulhon [29] and Alcubierre[30].Finally, note that as expected, the ADM formulation does not provide any equa-tions for the lapse function, α, or the shift vector, βi, as they represent the freedomof coordinate choice in General Relativity and can be set arbitrarily. However, aswe will discuss further in Sec. 1.7, a stable and long term numerical integrationof any 3+1 formulation of Einsteins’ equation is highly sensitive to the choice ofcoordinates.201.6. 3+1 Formulations of Einstein’s Equations1.6.2 Recasting of ADM Equations: BSSN FormulationIn principle, the standard ADM system, (1.41,1.42), can be used to evolve the 3-metric γij and extrinsic curvature Kij . This method is known as a free evolution,or an unconstrained evolution scheme since the Hamiltonian and momentum con-straints are only solved at the initial time. Another approach is to combine some orall of the constraint equations to determine some of the geometric variables at eachtime. Such methods are known as partially or fully constrained evolution schemes.It turns out that most unconstrained simulations using the ADM equations, es-pecially simulations of gravitational waves, are unstable. In particular, as we willdiscuss further in Chapter 4, in the case of gravitational waves it appears that theADM formulation is not even capable of evolving spacetime dynamics in the weakfield limit [28]. This can be traced back to the fact that the ADM equations are onlyweakly hyperbolic (for an extensive discussion on the notion of hyperbolicity andnumerical relativity see: [28]) which makes them unsuitable for numerical simula-tions. This observation provided the original motivation for Shibata and Nakamura[31] to recast the ADM equations in a way that is more applicable for numericalcalculations. This recasting was later revisited by Baumgarte and Shapiro [32], andis now known as the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation.The BSSN equations have become immensely popular in the numerical relativitycommunity due to the fact that they are strongly hyperbolic (and are free evolu-tion scheme). This property allows for successful long-time simulations of variousstrong gravity scenarios, most notably the simulations of compact binaries, includingextraction of the gravitational wave-form [13, 14].The BSSN formulation has 3 key features that differentiate it from the ADMformalism. In the following, we outline the corresponding changes that are made tothe ADM formulation that make the resulting equations strongly hyperbolic. Themanipulations of the equations in each step is straightforward, if sometimes lengthy,and we refer the reader to [28] for full details.211.6. 3+1 Formulations of Einstein’s EquationsI) Conformal rescalingIn the BSSN formulation, we split the evolution equations (1.41,1.42) of theADM variables {γij ,Kij} into the evolution of their overall scale and the evolutionof their “scale free” part13. This is done by a conformal re-scaling of these variables.First one defines the conformal factor, eφ, such that:e12φ ≡ γ , (1.43)in which γ is the determinant of the 3-metric γij . Then, the conformal metric γ˜ijdefined as:γij ≡ e4φγ˜ij ⇒ γ˜ = 1 , (1.44)has determinant 1. In addition, the conformally re-scaled extrinsic curvature A˜ijdefined through:Kij ≡ e4φA˜ij + 13γijK ⇒ Trace(A˜ij) = γ˜ijA˜ij = 0 , (1.45)is trace-free, i.e. scale free14. By fixing the determinant of γ˜ij and the trace of A˜ij ,the evolution equation for the 3-metric γij (1.41) splits into two equations for theoverall scale of the metric, φ, and the conformal metric γ˜ij. Similarly the evolutionequation for the extrinsic curvature Kij (1.42) splits into two equations for the traceof the extrinsic curvature, K, (i.e the overall scale of the tensor) and the trace-freepart A˜ij . The equations for φ and K can be derived by contraction of the twoevolution equations (1.41,1.42):(∂t −Lβ)(6φ) = (∂t −Lβ) ln γ1/2 = −αK , (1.46)13This is mainly motivated (heuristically) to separate the gravitational radiation part of thedynamics, encoded in the scale free part of the metric, and the overall gravitational field strengththat is approximately determined by the overall scale of the metric, and its main contribution comesfrom the matter distribution.14Note that the trace, K, is the “conjugate” scale to the determinant, γ. This will become clearshortly in the relation between the evolution of γ and K, i.e. Eq. 3+1 Formulations of Einstein’s Equations(∂t −Lβ)K = −D2α+ α (R− 4π(3ρ− S)) , (1.47)where we have used the definition of the conformal factor eφ (1.43). Here D2 denotesthe operator γijDiDj .The evolution equation for the scale free parts γ˜ij and A˜ij are then given by thescale free parts of the right hand side of equations (1.41,1.42):(∂t −Lβ)γ˜ij = −2αA˜ij , (1.48)(∂t −Lβ)A˜ij = e−4φ [−DiDjα+ α(Rij − 8πSij)]TF+ α(KA˜ij − 2A˜ilA˜lj) , (1.49)where the superscript TF denotes the trace-free part with respect to the 3-metric.Specifically, for any rank-2 tensor Xij we have:XTFij ≡ Xij −13(γklXkl)γij (1.50)II) Absorbing the Mixed DerivativesThe next step is to eliminate the mixed spatial derivatives appearing in the3-Ricci tensor in (1.49) by defining a conformal connection function:Γ˜k = γ˜ijΓ˜kij , (1.51)as a new dynamical variable. That is, Γ˜k, is to be evolved rather than being com-puted in terms of γ˜ij . The evolution equation for the conformal connection can bederived from its definition and the time evolution of the conformal metric (1.48),231.6. 3+1 Formulations of Einstein’s Equationsyielding:(∂t −Lβ)Γ˜i = −2A˜ij∂jα− 2α∂jA˜ij + 13γ˜li∂l∂jβj + γ˜lj∂j∂lβi . (1.52)Using the conformal re-scaling of the 3-metric, and the definition of the conformalconnection, Γ˜i, the 3-Ricci tensor is divided into two parts:Rij = Rφij + R˜ij . (1.53)The first term, Rφ, is associated with the overall conformal factor and is given byRφij = −2D˜iD˜jφ− 2γ˜ijD˜kD˜kφ+ 4D˜iφD˜jφ− 4γ˜ijD˜kφD˜kφ , (1.54)where D˜i is the covariant derivative associated with the conformal metric γ˜ij. Thesecond term in (1.53), R˜ij, is the 3-Ricci tensor associated with the conformal metricγ˜ij :R˜ij = −12γ˜lm∂m∂lγ˜ij + γ˜k(i∂j)Γ˜k + Γ˜kΓ˜(ij)k + γ˜lm(2Γ˜kl(iΓ˜j)km + Γ˜kimΓ˜klj). (1.55)where the conformal factor Γ˜k should be substituted by its evolved value via theevolution equation (1.52). As can be seen in (1.55), this process eliminated themixed spatial derivative terms (which could spoil the hyperbolicity) in R˜ij By thissubstitution, the principle part of the conformal 3-Ricci tensor is γ˜lm∂m∂lγ˜ij, whichis a wave-like spatial derivative operator.III) Adding the Constraints to the Evolution EquationsFinally, in the BSSN formulation, both the Hamiltonian and momentum con-straints are added to the evolution equations, creating a natural constraint dampingfeature to the free evolution equations. The Hamiltonian constraint is added tothe evolution equation of the extrinsic curvature with coefficient −α, (the lapse241.6. 3+1 Formulations of Einstein’s Equationsfunction):(∂t −Lβ)K = −D2α+ α (R− 4π(3ρ− S))− αH , (1.56)and the momentum constraint is added to the evolution of the conformal connection,with a coefficient 2α 15:(∂t −Lβ)Γ˜i = −2A˜ij∂jα− 2α∂jA˜ij + 13γ˜li∂l∂jβj + γ˜lj∂j∂lβi + 2αMi . (1.57)Adding the constraints 16 and expanding the Lie derivatives 17 results in the finalexplicit form of the BSSN equations summarized as following:Summary of BSSN equations:∂tφ = −16αK + βi∂iφ+16∂iβi , (1.58)∂tγ˜ij = −2αA˜ij + βk∂kγ˜ij + γ˜ik∂jβk + γ˜kj∂iβk − 23γ˜ij∂kβk , (1.59)∂tK = −γijDjDiα+ α(A˜ijA˜ij + 13K2) + 4πα(ρ+ S) + βi∂iK , (1.60)∂tA˜ij = e−4φ [−DiDjα+ α(Rij − 8πSij)]TF+ α(KA˜ij − 2A˜ilA˜lj)+ βk∂kA˜ij + A˜ik∂jβk + A˜kj∂iβk − 23A˜ij∂kβk , (1.61)15The choice can be ξα where ξ must be greater than 1/2, but otherwise arbitrary. Choosingξ = 2 leads to simpler equations.16and some algebra to express constraints equations in terms of the BSSN variables.17One should take into account that γ˜ij and A˜ij are tensor densities, φ is not a true scalar, buta scalar density related to the determinant of the metric, and similarly Γ˜i is a vector density. See[28].251.6. 3+1 Formulations of Einstein’s Equations∂tΓ˜i = −2A˜ij∂jα+ 2α(Γ˜ijkA˜kj − 23γ˜ij∂jK + 6A˜ij∂jφ− 8πγ˜ijSj)+ βj∂jΓ˜i − Γ˜j∂jβi + 23Γ˜i∂jβj +13γ˜li∂l∂jβj + γ˜lj∂j∂lβi . (1.62)In terms of the BSSN variables, the constraint equations are given by:H ≡ γ˜ijD˜iD˜jeφ − eφ8R˜+e5φ8A˜ijA˜ij − e5φ12K2 + 2πe5φρ = 0 , (1.63)Mi ≡ D˜j(e6φA˜ji)− 23e6φD˜iK − 8πe6φSi = 0 . (1.64)The above BSSN equations yield a well-posed Cauchy problem for Einstein’s equa-tions. If the initial data:(φ, γ˜ij ,K, A˜ij , ρ, Si, Sij)|t=0 , (1.65)satisfies the Hamiltonian and momentum constraints18 (1.63,1.64), and Γ˜i is initial-ized by (1.51), then experience has shown that the system can be evolved via theBSSN evolution equations (1.58-1.62) stably and for a long time. The BSSN for-mulation does not require use of the constraint equations (apart from determiningthe initial data), and therefore is a fully evolutionary formulation of the Einstein’sequations.18Numerically, the initial data satisfies these equations only approximately. To have a consistentset of initial data and evolution system, the error in the initial data is expected to be as small as thetruncation error (the error due to the finite difference approximation of the continuum equations)of the evolution scheme.261.7. Coordinate Choices1.7 Coordinate ChoicesThe coordinate freedom in General Relativity is encoded in the choice of the lapsefunction α and the shift vector βi. Intuitively, the lapse function defines the shapeof the embedding of the 3-dimensional hypersurfaces Σt within the spacetime, whilethe shift vector—that is defined as the deviation of the spatial coordinate from thenormal direction to the hypersurfaces (Fig. 1.1)—determines the dynamics of thespatial coordinates relative to normal propagation. Choosing good forms for thesefunctions has been a significant part of the research work in the field of numericalrelativity. In fact, finding a dynamical coordinate choice (a notion that will be intro-duced shortly) that allows long-time and stable evolution of a strongly gravitatingsystem is not a trivial task.Two scenarios that pose particular challenges for the coordinate choices are theevolution of a spacetime that contains a black hole or the evolution of a spacetimecontaining matter collapsing to a black hole. As can be seen from the Schwarzschildsolution (1.14), a specific coordinate choice can be singular (not necessarily only atthe event horizon) where the singularity is not physical and can be removed by acoordinate transformation. A more challenging case is a physical singularity (pointof infinite curvature) that can form, for example, at the center of symmetry in acollapse scenario. Obviously, a numerical code cannot continue execution if it en-counters such a singular point during a simulation. A particular type of coordinatechoice is necessary to appropriately deal with this scenario and is known as singular-ity avoiding. Intuitively, the lapse function is chosen so that it slows the evolutionin the vicinity of a singularity.Even if the spacetime does not contain a black hole or is not evolving towardthe formation of one, the coordinate choice can still require special attention, sincea (non-physical) coordinate singularity can form. This can be demonstrated usingthe simplest gauge choice:α = 1 , βi = 0 , (1.66)271.7. Coordinate Choiceswhich are known as Gaussian-normal coordinates, and which is a particular case ofa coordinate system that incorporates geodesic slicing. Due to the vanishing shiftvector, the coordinate time vector coincides with the normal vector, and due to thechoice α = 1, the observers moving along the direction normal to the hypersurfacesmeasure coordinate time which is equal to proper time. The acceleration of thenormal observers is given by ab = Dbα = 0; hence xi = cons. trajectories are indeedgeodesics. In this gauge choice, a coordinate pathology can develop quickly as canbe seen from the evolution equations of the extrinsic curvature and the determinantof the 3-metric. In the Gaussian-normal system the equations (1.46) and (1.60) invacuum are given by:∂t ln γ1/2 = −K , (1.67)∂tK = A˜ijA˜ij +13K2 > 0 . (1.68)From the second equation (1.68) K grows monotonically in time, while from (1.67)the volume element of the 3-surfaces decays as the extrinsic curvature becomesunbounded (due to the K2 term in (1.68)). Intuitively, this behaviour can be seenfrom the definition of the extrinsic curvature as the negative of the expansion of thenormal direction—which in geodesic slicing is equal to the convergence (negativeexpansion) of the geodesics. The development of such coordinate singularities isnot surprising as the geodesics can focus toward each other—for example from agravitational radiation passing through a region of spacetime—and nothing preventsthem from crossing.Motivated by this geometric observation, one well-known coordinate choice isdeveloped by imposing a condition that prevents the normals from converging. Fromthe definition of the extrinsic curvature, this can be achieved by requiringK = 0 . (1.69)281.7. Coordinate ChoicesThis coordinate choice is known as the maximal slicing condition. Provided K = 0at t = 0, the evolution equation of the extrinsic curvature (1.60) requires0 = −γijDjDiα+ α(KijKij + 4π(ρ+ s)) , (1.70)for all subsequent t in order for the maximal slicing condition to hold. This lastequation can be solved as an elliptic equation for the lapse function.Imposing a specific constraint, such as (1.69), on some of the variables of theADM formulation, is one of the common approaches to define coordinates, and willtypically create constraint-type equations for the gauge variables α and βi. For theshift vector, one example of this type of conditions is the quasi-isotropic choice whereone requires the 3-metric to remain diagonal. Assuming the 3-metric is diagonal att = 0, this can be accomplished by requiring that the time derivative of the off-diagonal 3-metric components be 0 for all t > 0. This demand then yields0 = −2αKij +Diβj +Djβi , i 6= j , (1.71)which can be solved as a boundary value problem for the shift vector. We note thatin general there are 3 degrees of freedom to choose the spatial coordinate on the3-hypersurfaces, which can be used to fix the 3 free variables in the shift vector.However, for certain symmetric metrics, conditions such as (1.71) may not result ina sufficient number of equations, and extra algebraic condition may be needed inconjunction with them to completely fix the shift vector [28].Another approach to specify the coordinate system, which is the main choicein Chapter 3 and 4 of this thesis, is by defining evolution equations for α and βi.The constraint-type gauge choices, such as maximal slicing, usually result in a set ofcomplex elliptic equations that needs to be solved on every time slice. Implementingan effective elliptic-solver for such equations is a challenging computational task. Incontrast, if one can prescribe evolution equations for the coordinate system, then it291.7. Coordinate Choicesis quite straight-forward to integrate them forward in time along with the evolutionequations for the dynamical variables.One particular dynamical slicing condition that has proven to be robust andwhich has the singularity avoidance property is the following∂tα = −2αK . (1.72)This coordinate choice is known as the 1+log19 condition. It was implemented inthe original BSSN work and now is considered its standard lapse choice. The 1+logcondition can be viewed as a “K-damping” evolution equation, in the sense that theevolution of the lapse effectively damps the growth of K in the coupled system ofPDEs for K and α. Overall, the 1+log gauge has been shown to mimic the maximalslicing condition (but requires much less computational effort to implement) and inparticular has the singularity avoidance property20.We now want to consider conditions for the shift vector that are similarly dy-namical. First, looking at the BSSN equations (1.58-1.62), we see that the systemsimplifies if the conformal connection Γ˜i can be set to zero. However such a condi-tion, if implemented exactly, will lead to an elliptic type equation for the shift (knownas Gamma-freezing) and may in fact spoil the hyperbolicity of the BSSN system.Taking a similar approach to 1+log slicing, which creates a natural K-damping sys-tem, as opposed to exactly enforcing K = 0, one can impose the following evolutionequation for the shift vector:∂tβi = µΓ˜i − ηβi . (1.73)This is known as the Gamma-driver shift condition. Here, µ and η are adjustable19The name “1+log” is due to the fact that equation (1.72) combined with (1.58) and zero shiftimplies: ∂tα− 12∂tφ = 0 which can be solved by assuming α = 1+ 12φ and using the definition ofφ (1.43) this condition is equivalent to: α = 1 + ln(γ)—hence the name: 1+log.20We also note that the maximal slicing condition cannot be used in the BSSN formulation, as itspoils the hyperbolicity of the equations, making the resulting system ill-posed.301.8. Overview of Numerical Techniques for Time Dependent Problemsnumerical parameters. µ is usually chosen to be 3/4 21, while η has the unit ofinverse of time, and since the time scale of the system is usually set by the totalmass of the system, η is chosen to be of order 1/M where M is the total massof the spacetime (ADM mass). Again, as with the 1+log choice, the Gamma-driver condition effectively creates a damping term and controls the value of theconformal factor Γ˜i. Together, the 1+log and Gamma-drive conditions have provento be successful in dealing with challenging problems in numerical relativity, suchas binary black hole coalescence. These two coordinate choices are used in Chapter3 and 4 of this thesis, where they are implemented to study type II critical collapsescenarios.1.8 Overview of Numerical Techniques for TimeDependent ProblemsWhen Einstein’s field equations are decomposed through a 3+1 formulation, theybecome a set of time dependent PDEs. A standard approach to numerically solvePDEs is the finite difference methods, where continuum functions are discretized:f(t, ~X)→ f(tn, xi, yj, · · · ) ≡ fnij··· ≡ fnI , (1.74)and differential operators are replaced with difference operators. For example:∂∂Xf(X)→(E + E−12∆X)(fI) ≡ fI+1 − fI−12∆X. (1.75)In (1.74) and (1.75) tn denotes discrete values of time tn = t0+n∆t, XI symbolizesa mesh along one of the spatial coordinates: XI = Xmin + I∆X (for example: XIcan be xi = xmin + i∆x), ∆X and ∆t are the step sizes of the discretization in t21This particular value has been found from purely numerical experiments and is known toperform well.311.8. Overview of Numerical Techniques for Time Dependent Problemsand X and E denotes the shift operator along X:E(fI) ≡ fI+1 , (1.76)E−1(fI) = fI−1 , (1.77)and is the fundamental operator that creates all the difference operators. Operationssimilar to (1.75) can be defined to replace time derivatives with finite differenceexpressions. The key assumption here is the smoothness of the function that allowsTaylor expansion. For instance, the Taylor expansion of the RHS of (1.75) yieldsfi+1 + fi−12∆x=dfdx+16d3fdx3(∆x)2+1120d5fdx5(∆x)4+· · · = dfdx+O(∆x2) ≈ dfdx, (1.78)where we are using big-O notation—O(∆x2) is a function that converges to zeroas fast as ∆x2. Neglecting this term, the LHS of 1.78 becomes a Finite DifferenceApproximation (FDA) to the differential operator d/dx. The O(∆x2) term is usuallyreferred as the truncation error of the FDA.In a nutshell, the discretization process converts a PDE to a finite differenceequation which is an algebraic equation that can be solved numerically on a com-puter. However, in practice, this process has several complications, including find-ing the FDA operators for derivatives with the correct accuracy, handling boundarypoints, initialization, developing testing facilities and generating solver routines. Wedeveloped a Maple based toolkit called FD that simplifies these steps while allowingfull control over the entire process while helping the user to focus on the underlyingphysical/mathematical phenomena described by the PDE. This toolkit is a set ofMaple procedures and definitions that provides a high level language to specify aPDE over a discretized numerical domain and solve it. It can compute the finitedifference approximation (FDA) equivalent of a PDE and generate low level lan-guage (Fortran) routines and C wrappers that evaluate the FDA expression or solveit for the dynamical (unknown) field. FD also allows a rapid prototyping work-flow321.9. Outline of the Thesisto create the diagnostic facilities used in finite difference methods, and generatesroutines that are parallel ready that can be used within a framework of a parallelcomputing infrastructure such as PAMR [33].The Appendix of this thesis, in large part, is the user manual for this software.It also discusses the details and key concepts of the finite difference method as wellas the mathematical notion of convergence and independent residual evaluators thatare used throughout this thesis as diagnostic tools. At any point, the materials inthe Appendix can be consulted as a pedagogical reference.1.9 Outline of the ThesisAs mentioned above, this thesis is concerned with both Type I and Type II criti-cal collapse. In Chapter 2, we study the Einstein-Vlasov model which describes aset of collision-less particles modeled as a phase-space distribution coupled to Ein-stein’s equations. This study is to address some of the inconsistencies in the typeI critical collapse studies of Einstein-Vlasov system, and in particular to focus onthe massless22 system and understand the role of the angular momentum. We di-rectly integrate the Vlasov equation in phase-space, i.e. we evolve the distributionf(t, r, pr, l2) where f is the density of the particles in phase-space, r is the radialcoordinate, pr is the radial momentum of the particles and l2 is the angular mo-mentum of the particles. The geometry and phase-space distribution are restrictedto spherical symmetry, however the dynamics of f in the phase-space is indeed a 3+ 1 computation, 3 phase-space dimensions (r, pr, l2) and 1 time coordinate. Themain finding in this section is the observation of type I critical behaviour and theuniversality of the time-scaling exponent in the massless system. In addition, wefind a family of static solutions to the massless system and show that they all canplay the role of type I critical solutions with similar time-scaling exponent whenthey are perturbed. This chapter also explains the numerical techniques and the22Particles are moving along the null geodesics of the spacetime.331.9. Outline of the Thesisnew finite-volume code we developed to solve the Vlasov part of the problem. Thechapter is identical to its published version [1].Chapter 3 and 4 are focused on type II critical collapse. In particular, we areinterested in applying a modified version of the BSSN formulation, known as Gener-alized BSSN, to type II critical solutions and ultimately to develop an axisymmetriccode that can be used in various type II critical phenomena studies. The ultimategoal is to extend the limited studies in the numerical relativity literature in typeII critical collapse and in particular to find an appropriate formulation and coordi-nates choice that allow generic type II critical phenomena studies among which thecritical collapse of pure gravitational waves is yet an unresolved problem.In Chapter 3, we begin with adopting the BSSN formulation in spherical sym-metry and apply the technique to the well-known problem of critical collapse of amassless scalar field. As will be discussed, the use of free evolution schemes anddynamical coordinate choices has not been successful in the past in resolving thediscrete self similarity of type II threshold solutions. The main result of Chapter3 is the first successful implementation of a hyperbolic formulation that is capableof evolving the spacetime dynamics sufficiently close to the critical solution to al-low observation of the characteristics of the DSS spacetime. Our results establish apotential route to extend the type II critical phenomena studies in axial symmetryusing the G-BSSN formulation. This chapter is also identical to its published version[2].Chapter 4 extends techniques we implemented in spherical case to axial symme-try. We describe the implementation of a new generalized BSSN axisymmetric codethat uses cylindrical coordinates and provide evidence confirming its robustness andaccuracy. The code can in principal be coupled to any matter sources. However,we demonstrate the performance of the code in the strong pure gravitational wavescontent, which has been historically the most challenging case. We evolve highlynon-linear gravitational waves in axial symmetry where the vacuum can collapse341.9. Outline of the Thesisto a black hole. Our primary calculations suggest that, again, generalized BSSNappears to be promising, and an extension of the work can shed more insight, andperhaps solve the as yet unresolved problem of critical collapse in pure gravitationalwaves.35Chapter 2Critical Collapse in theSpherically SymmetricEinstein - Vlasov Model2.1 IntroductionIn this paper23 we report results from an investigation of critical collapse in thespherically symmetric Einstein-Vlasov system, which describes the interaction ofcollisionless matter with a general relativistic gravitational field. After more thantwo decades of study, the field of black hole critical phenomena has matured andalthough we present a brief overview below, we assume that the reader is at leastsomewhat familiar with the key concepts and results in the subject: those who arenot can consult comprehensive review articles [5, 23].We recall that critical phenomena can be identified in a given model by consid-ering dynamical evolution of initial data that is characterized by a parameter, p,such that for sufficiently small p the gravitational interaction remains weak and thematter (or gravitational energy in the vacuum case) typically disperses, while forsufficiently large p a black hole forms. By tuning p between these limits we isolatea critical parameter value p⋆ that generates a solution representing the threshold ofblack hole formation for the particular family of initial data. The behaviour that23This chapter is published in: Akbarian. A. and Choptuik M. W. “Critical collapse in thespherically-symmetric Einstein-Vlasov model. Phys. Rev. D90, 104023 (2014).362.1. Introductionarises in the near-critical regime p→ p⋆ constitutes what is meant by black hole crit-ical phenomena. Depending on the particulars of the model, these phenomena willcomprise one or more of the following: 1) existence of a special solution at criticalitywith possible universality with respect to the parameterization of the initial data,2) symmetry of the critical solution beyond any imposed in the model itself and3) scaling of dimensionful physical quantities as a function of |p − p⋆|, with scalingexponents which may also be universal in the sense given above. These propertiescan largely be explained by observing that a critical solution has a single unstablemode in perturbation theory, whose associated eigenvalue (Lyapunov exponent) canbe immediately related to the empirically measured scaling exponent.For the most part, the critical transitions that have been observed to date fallinto two classes that are dubbed type I and type II in analogy with first and secondorder phase transitions, respectively, in statistical mechanical systems, and wherethe behaviour of the black hole mass plays the role of an order parameter. A typeI transition is characterized by a static or periodic critical solution, with a scalinglawτ = −σ ln |p− p⋆| . (2.1)Here, τ is the lifetime of the near-critical configuration—the amount of time that thedynamical configuration is closely approximated by the precisely critical solution—and the scaling exponent, σ, is the reciprocal of the Lyapunov exponent, λ, asso-ciated with the solution’s single unstable mode. In this case the black hole massis finite at threshold since when the marginally stable static or periodic solutioncollapses, most of its mass-energy will end up inside the horizon.Previous studies [34–37] have strongly suggested that the critical behaviour inthe Einstein-Vlasov model is generically type I and our current results bear this out.So far as we know, type II collapse, where the critical solution is self similar and theblack hole mass is infinitesimal at threshold, is not relevant to the model and willnot be considered here.372.1. IntroductionIn the Einstein-Vlasov system the matter content of spacetime is specified by adensity function f(t, xi, pj) in phase space whose evolution is given by the Vlasovequation, while the geometry is governed by the Einstein equations. Numericalstudies of the model have a long history, dating back to the work by Shapiro andTeukolsky, both in spherical symmetry [38–40] and axisymmetry [41, 42]. Investiga-tion of critical collapse in the spherically symmetric sector was initiated by Rein etal [34] who observed finite black hole masses at threshold for all families considered.Subsequent work by Olabarrieta and Choptuik [35] corroborated these findings andadditionally provided evidence that the threshold solutions were static with lifetimescaling of the form (2.1). Moreover, there were some indications in this latter studythat there might be a universal critical solution and associated scaling exponent.More recently, Andre´asson and Rein have carried out a comprehensive study ofprecisely static solutions of the model, concentrating on their stability both generallyand in the context of critical phenomena [37, 43]. Many of their observations andresults are pertinent to our current investigation. First, they point out that staticsolutions can be constructed via a specific ansatz for the distribution function thatis discussed in Sec. 2.3. Second, using this ansatz they construct parameterizedsequences of static solutions, and, following astrophysical practice, characterize thesolutions by their central redshifts and binding energies. Third, they present strongevidence that a maximum in the binding energy along a sequence signals an onsetof instability and that at least some of the configurations that lie along an unstablebranch can act as type I solutions in the critical collapse context. This immediatelyestablishes that there can not be universality in the model. Fourth, and finally, theyshow that dispersal is not the only stable end state of sub-critical collapse, but thatrelaxation to a bound state is also possible, contingent on the sign of the bindingenergy. Overall, the picture of critical behaviour that emerges very much parallelsthat which is observed for type I transitions in the perfect-fluid and massive-scalarcases [44–51].382.1. IntroductionAll of the work reviewed above used a non-zero particle mass. However, themassless case can also be considered and the current research is largely aimed atexploration of that sector. Additionally, we attempt to address some issues thatremained open following Andre´asson and Rein’s work, including whether there isany explanation for the indications of universality seen in [35]. We note that for themassless model Martin-Garcia and Gundlach [52] considered the possibility of theexistence of one-mode unstable self similar configurations that could serve as type IIcritical solutions. Interestingly, they concluded that since there are infinitely manymatter configurations that give rise to any given static spacetime, any unstable so-lution must have an infinite number of unstable modes. Their argument also appliesto the static case, which then suggests that there should be no type I behaviour inthe model either.In spherical symmetry the Vlasov equation is a PDE in time and three phasespace dimensions.24 Thus, direct numerical solution is costly and this fact motivatedthe use of particle-based algorithms in all previous studies excepting [36]. However, akey deficiency of particle approaches is that the results develop a stochastic characteron a short time scale. This leads to poor convergence properties relative to a directmethod, namely an error that is only O(1/√N), where N is the number of particles.With the substantial increase in computational resources over time, direct solutiontechniques have become feasible and about a decade ago Stevenson [36] implementeda finite-volume solver for the Vlasov PDE for the case that all particles have thesame angular momentum. The code that we have developed is largely a continuationof his effort and produces results that have well-behaved convergence properties asa function of the mesh spacing.Our numerical studies are based on two types of initial data. The first, which weterm generic, is characterized by a relatively arbitrary functional form for f(0, xi, pj).The second, which we call near static, is based on perturbations about some precisely24The 3 phase space dimensions are: radial direction r, radial momentum pr and the angularmomentum l (see Sec. 2.2.2).392.2. Equations of Motionstatic solution that is constructed from the ansatz described in Sec. 2.3. We performexperiments using initial conditions of the first type for both massless and massiveparticles, but restrict attention to the massless sector for our near-static studies.Aiming to unearth as much phenomenology as possible, as well as to explore theissue of universality, we have attempted to broadly survey the possibilities for thespecific form of the initial distribution function in all three sets of experiments.The remainder of the paper is structured as follows. The next section describesthe equations of motion for the model while Sec. 2.3 discusses the construction ofstatic solutions from the ansatz mentioned previously. Sec. 2.4 details our numericalapproach, including code validation. Sec. 2.5 is devoted to the main results fromour study and we conclude with a summary and discussion in Sec. 2.6. We haveadopted units in which G = c = 1.2.2 Equations of MotionA configuration of a system of particles can be described by the phase space den-sity, f(t, xi, pj), also known as the distribution function, where xi and pj are theparticles’ spatial positions and 3-momenta, respectively. In the Einstein-Vlasov sys-tem particles interact only through gravity. Consequently, the particles move ongeodesics of the spacetime along which the density function is conserved25:Df(t, xj, pj)dτ= 0 . (2.2)Here, τ is the proper time of the particle and D/dτ is the Liouville operator:Ddτ≡ dxµdτ∂∂xµ+dpjdτ∂∂pj. (2.3)25This can be viewed as the conservation of the particles number in the volume element of thephase space (co-moving with the particles) since we assume no collision between the particles. Anintroduction to Einstein-Vlasov is given in [53] and an extensive textbook in relativistic Boltzmannequation is [54].402.2. Equations of MotionUsing the geodesic equationvµ∂µpν − vµΓλµνpλ = 0 , (2.4)where vµ is the particle 4-velocity, the Vlasov equation can be written aspµ∂f∂xµ+ pνpλΓλνj∂f∂pj= 0 . (2.5)The energy momentum tensor of the system is given by integrating over themomentum of the particles:Tµν(t, xi) =∫pµpνmf(t, xi, pj)dVpj , (2.6)where m is the particle mass. Equations (2.5) and (2.6), together with Einstein’sequationsGµν = 8πTµν , (2.7)govern the evolution of the Einstein-Vlasov system. These equations, restricted tospherical symmetry by requiring f(t, xi, pj) = f(t, R(xi), R(pj)), R ∈ SO(3) is thesystem we study numerically.2.2.1 Coordinate Choice and Equations for Metric ComponentsWe adopt polar-areal coordinates (t, r) in which the spherically-symmetric metrictakes the formds2 = −α(t, r)2dt2 + a(t, r)2dr2 + r2dθ2 + r2 sin2 θdφ2 . (2.8)The radial metric function a(t, r) can be determined from either the Hamiltonianconstraint,a′a=1− a22r− ra228πT t t , (2.9)412.2. Equations of Motionwhere ′ ≡ ∂/∂r, or from the momentum constraint,a˙a=ra228πT r t , (2.10)with ˙ ≡ ∂/∂t. The lapse function α(t, r) is fixed by the polar slicing-conditionα′α=a2 − 12r+ra228πT r r . (2.11)Equation (2.9) is solved subject to the boundary condition,a(t, 0) = 1 , (2.12)which follows from the demand of elementary flatness at the origin. For the lapsewe setα(t, rmax) =1a(t, rmax), (2.13)where rmax is the location of the outer boundary of the computational domain, sothat coordinate and proper time coincide at infinity.The θθ component of Einstein’s equation yields an additional redundant equa-tion, and we use the degree to which it is satisfied as a check of our numericalresults.2.2.2 The Energy Momentum TensorAs noted above, for a given distribution function, f(t, xi, pj), the stress tensor iscomputed from the momentum-space integral (2.6). With our choice of metric thevolume element is given bydVpj =md3pjp0√|g| =mdprdpθdpφp0αar2 sin θ. (2.14)422.2. Equations of MotionTo impose spherical symmetry we require the distribution function to be uniform inall possible angular directions. This condition can be conveniently implemented bytransforming to variables l2 and ψ given byl2 ≡ p2θ +p2φsin2 θ, (2.15)ψ ≡ tan−1(pθ sin θpφ), (2.16)where l is the angular momentum of the particles. Spherical symmetry is thenachieved by demanding that f(t, xi, pr, l2, ψ) ≡ f(t, r, θ, φ, pr, l2, ψ) = f(t, r, pr, l2).The volume element in the new variables isdVpj =mdprdl2dψ2ap¯tr2, (2.17)wherep¯t ≡ αp0 =√m2 +p2ra2+l2r2. (2.18)Integrating over ψ, the components of the energy momentum tensor are givenby:T t t =−πar2∫∫p¯tfdprdl2 , (2.19)T r r =πa3r2∫∫p2rp¯tfdprdl2 , (2.20)T r t =−παa3r2∫∫prfdprdl2 , (2.21)T θθ =−π2ar4∫∫l2fp¯tdprdl2 . (2.22)432.3. Static Solutions2.2.3 Evolution of the Distribution FunctionHaving imposed spherical symmetry the Vlasov equation (2.5) can be written aspt∂f∂t+ pr∂f∂r+(α′p2tα3+a′p2ra3pt+l2r3)∂f∂pr= 0 . (2.23)By definingg ≡ αprα2p¯t=∂H∂pr, (2.24)h ≡ −α′p¯t + αa′p2ra3p¯t+αl2r3p¯t= −∂H∂r, (2.25)where H is the Hamiltonian,H ≡ α√m2 + (pr/a)2 + (l/r)2 , (2.26)equation (2.23) can be cast as a conservation law:∂f∂t− {H, f} = ∂f∂t+∂(gf)∂r+∂(hf)∂pr= 0 . (2.27)This form of the Vlasov equation facilitates the use of finite-volume techniques inour numerical treatment of the problem.2.3 Static SolutionsSpherically symmetric static solutions of the Vlasov equation can be generated bysimply requiring that the distribution function at the initial time take the formf(0, r, pr, l2) = Φ(E, l), whereE ≡ α√m2 + (pr/a)2 + (l/r)2 (2.28)442.3. Static Solutionsis the energy of the particles and, again, l is the angular momentum parameter [55].Indeed, since E and l are both conserved along particle geodesics in spherical sym-metry, any distribution function of this form remains unchanged as the particlesmove and the Vlasov equation is automatically satisfied.Explicit construction of the static spacetime resulting from a given choice ofΦ(E, l) requires that the metric functions α and a be determined self-consistently.To that end we can write (2.9) and (2.11) as−2r∂r ln a+ 1a2− 1 = 8πr2T tt(r;α,Φ) , (2.29)2r∂r lnα+ 1a2− 1 = 8πr2T rr(r;α,Φ) , (2.30)whereT tt(r;α,Φ) = −πr2∫∫p¯tΦ(E(α, r, w, l), l)dw dl2 , (2.31)T rr(r;α,Φ) =πr2∫∫w2p¯tΦ(E(α, r, w, l), l)dw dl2 , (2.32)w =pra, (2.33)p¯t =√m2 + w2 + (l/r)2 , (2.34)E = α√m2 + w2 + (l/r)2 . (2.35)Given a functional form for Φ(E, l), we can integrate the equations for α(r) and a(r)from r = 0 outward, subject to the boundary conditions (2.12)-(2.13). Physically,we also want the particle distribution resulting from a given Φ(E, l) to have compactsupport in phase space and finite total mass. As shown in [56], these conditions can452.4. Numerical Techniquesbe satisfied by introducing a maximum (cut-off) energy, E0, so thatΦ(E, l) = φ(E/E0)Θ(E0 − E)F (l) , (2.36)where Θ is the unit step function. In Sec. 2.5.2 we construct static solutions basedon this ansatz and then investigate their relationship to critical behaviour in themodel.2.4 Numerical TechniquesIn this section we summarize our numerical approach for constructing approximatesolutions of the equations of motion and the various tests we have performed toestablish the correctness and accuracy of our implementation.2.4.1 Evolution SchemeAs previously mentioned, we treat the matter evolution by a direct discretization ofthe multidimensional Vlasov equation. Relative to the particle methods adopted inmost previous studies of the Einstein-Vlasov system, this has the advantage that ournumerical solutions have superior convergence properties. In particular, in contrastto the particle approach, there is no stochastic component of the solution error. Thisin turn leads to improved confidence in our identification of key aspects of the criticalphenomena exhibited in the model, including 1) evidence that the threshold solutionsare static and 2) the scaling exponents associated with the critical configurations.As also noted above, the Vlasov equation can be expressed in conservation formand is thus amenable to solution using finite-volume methods. These techniques,which are used extensively in fluid dynamics, for example, are well known for theirability to accurately resolve sharp features—including discontinuities—that oftenappear in the solution of conservation laws. In our case, evolutions of the distribu-tion function generically exhibit significant mixing and steep gradients; moreover,462.4. Numerical Techniquessome of our computations involve initial data which is not smooth in phase space.26The finite-volume strategy is thus natural for our purposes. We sketch our specificapproach by considering the general form of a conservation equation for a quantityq(t, x, y):∂q(t, x, y)∂t+∂Fx(q)∂x+∂Fy(q)∂y= 0 , (2.37)where Fx(q) and Fy(q) are the fluxes in the x and y directions. We follow the usualfinite volume approach (see [57] for example) by dividing the computational domaininto Nx × Ny cells of uniform size ∆x × ∆y as shown in Fig. 2.1, and define theaverage value of the unknown q over the cell Cij byQnij =1∆x∆y∫∫Cijq(tn, x, y)dxdy . (2.38)Here the superscript n labels the discrete time, tn ≡ n∆t. We then rewrite (2.37)in integral form:∂Q∂t= − 1∆x∆y(∫EFx(q)dy −∫WFx(q)dy)− 1∆x∆y(∫NFy(q)dx−∫SFy(q)dx), (2.39)where the subscripts E, W, N and S denote the east, west, north and south bound-aries, respectively, of the cell Cij. Applying a time-discretization to this last expres-sion yields an equation that can be used to advance the cell average in time:Qn+1ij = Qnij −∆t∆x([Fx]ni+1/2 − [Fx]ni−1/2)−∆t∆y([Fy]nj+1/2 − [Fy]nj−1/2). (2.40)Here the average fluxes at the boundaries, [Fx]ni+1/2 etc. are calculated using a Roesolver [57]. We note that our calculations are always performed on meshes that26See for example the choice of b = 1 in Eq. (2.70). The radial density distribution of the particlesin this choice is not smooth as can be seen in the right panel in Fig. 2.10 (labeled as b = 1).472.4. Numerical Techniquesx[ ]−1/2niFjFxij−1j+1yF[ ]j +1/2[ ]i +1/2Q nijnnn−1/2jy]F[ i+1i−1Figure 2.1: A portion of the discretized computation domain used in our finitevolume code. The dashed lines delineate one finite volume cell. The cell-centredaverage value of the density, Qnij is defined on the grid points marked with filledcircles while the fluxes, [Fx]ni−1/2, [Fx]ni+1/2, etc. are computed at points denotedwith dashed circles and which lie on cell boundaries. As described in more detailin the text, Qnij is updated using the difference of the outgoing and ingoing fluxesthrough the cell boundaries.482.4. Numerical Techniquesare uniform in each coordinate direction, and that when we change resolution—toperform a convergence test for example—each mesh spacing is changed by the samefactor. Thus, our discretization is fundamentally characterized by a single scale, h.Our specific finite volume approach is based on O(h2) approximations. However,the nature of the flux calculations—which are designed to inhibit the developmentof spurious oscillations—means that the scheme is only O(h) in the vicinity of anylocal extrema in the solution.The metric variables α and a, which need only be defined on a mesh in the rdirection, are computed from O(h2) finite difference approximations of the Hamil-tonian and slicing equations, (2.9) and (2.11). Since the equations for the matterand geometry are fully coupled—i.e. α and a appear in the flux computations,and f is needed for the calculation of the source terms for α and a—some care isneeded to construct a scheme which is fully O(h2) accurate (modulo the degrada-tion of convergence near extremal solution values just noted). In practice, we usean O(∆t2) = O(h2) Runge-Kutta scheme for the time stepping, which necessitatescomputation of auxiliary quantities at the half time step tn+1/2 = tn + ∆t/2. Ouroverall scheme that advances the solution from tn to tn+1, and which does haveO(h2) truncation error, is:1. Compute fn+1/2 from (2.40) using the fluxes Fn.2. Compute a˜n+1/2 from (2.10) with source [T rt]n.3. Compute[T tt]n+1/2and [T rr]n+1/2 from (2.19)–(2.20) using a˜n+1/2.4. Compute an+1/2 and αn+1/2 from (2.9) and (2.11) with sources[T tt]n+1/2and[T rr]n+1/2.5. Compute [T rt]n+1/2 from (2.21).6. Compute fluxes Fn+1/2x and Fn+1/2y using an+1/2 and αn+1/2.7. Compute fn+1 from (2.40) and the half-step fluxes Fn+1/2.492.4. Numerical Techniques8. Compute a˜n+1 from (2.10) with source [T rt]n+1/2.9. Compute[T tt]n+1and [T rr]n+1 from (2.19) and (2.20) using a˜n+1.10. Compute an+1 and αn+1 from (2.9) and (2.11) using sources[T tt]n+1and[T rr]n+1.11. Compute [T rt]n+1 from (2.21).12. Compute fluxes Fn+1x and Fn+1y using an+1 and αn+1.13. One time step complete; start next time step.To facilitate the use of large grid sizes, as well as to speed up the simulations,we parallelize the computations for the evolution of the distribution function andthe calculation of the energy-momentum tensor components using the PAMR 27package [33]. On the other hand, the calculation of the metric components, whichhas negligible cost relative to the updates of f and T µν , is performed on a singleprocessor. The new values of the metric functions are then broadcast to the otherCPUs.2.4.2 Initial DataIn spherical symmetry the gravitational field has no dynamics beyond that generatedby the matter content, so initial conditions for our model are completely fixed by thespecification of the initial-time particle distribution function, f(0, r, pr, l2). However,the Einstein equations (2.9)–(2.11) must also be satisfied at the initial time and,through the definition (2.18) for p¯t, a appears within the integrands for the stresstensor components. To determine all requisite initial values consistently we thereforeuse the following iterative scheme:27Parallel Adaptive Mesh Refinement: a software developed by Fran Pretorius for parallelizingtime dependent PDE solver codes and applying AMR algorithm.502.4. Numerical Techniques1. Initialize the distribution function, f(0, r, pr, l2), to a localized function onphase space.2. Initialize the geometry to flat spacetime.3. Calculate the energy momentum tensor using the current geometry.4. Calculate the geometry using the current energy momentum tensor.5. Iterate over the matter and geometry calculations until a certain tolerance isachieved.In practice we find that this algorithm converges in a few iterations.As discussed in Sec. 2.5.2, when we study static initial data we first specifyΦ(E, l) and then integrate (2.29)–(2.30) outward. We note that the form of Φ(E, l)that we choose,Φ(E, l) = φ(E/E0)Θ(E0 − E)F (l) , (2.41)results in equations that are invariant under the transformation:α → kα , (2.42)E0 → kE0 . (2.43)We can thus first integrate the slicing condition (2.30) subject to the boundarycondition, α(0, 0) = Λ, with Λ < 1 but otherwise arbitrary, and then linearly rescaleα(0, r) so that α(0, rmax) = 1/a(0, rmax). The central redshift of the configuration,Zc, which we use in our analysis below, is then given byZc ≡ 1α(0, 0)− 1 , (2.44)where α(0, 0) is now the rescaled value. It is important to emphasize that differ-ent choices for Λ result in distinct solutions, so that irrespective of any adjustable512.4. Numerical Techniquesparameters that may appear in the specification of φ, equation (2.41) will alwaysimplicitly define an entire family of static configurations.2.4.3 Diagnostic Quantities and Numerical TestsWe have validated our implementations of the algorithms described above usinga standard convergence testing methodology that examines the behaviour of thenumerical solutions as a function of the mesh spacing, h, keeping the initial datafixed. This section summarizes the tests we perform—which involve derived quan-tities that should be conserved in the continuum limit as well as the full solutionsthemselves—and presents results from their application to a representative initialdata set using three scales of discretization, h, h/2 and h/4.Conserved QuantitiesThe mass aspect function, m(t, r), is given bym(t, r) =r2(1− 1a2(t, r)), (2.45)and measures the amount of mass contained within radius r at time t. Its value atspatial infinityM ≡ m(t,∞) , (2.46)is the conserved ADM mass. Alternatively, M can be computed usingM =∫ ∞0ρ4πr2dr , (2.47)ρ = nµnνTµν , (2.48)where nµ is the unit timelike vector normal to the spatial slices. In developing ourcode we computed mass estimates based on both of these expressions, but the resultspresented here and in the remainder of the paper use (2.46) exclusively. Fig. 2.2(c)522.4. Numerical Techniquesgraphs deviations of M relative to its time-averaged mean value 〈M〉 for the threecomputations performed with mesh scales h, h/2 and h/4. As noted in the caption,the values ofM−〈M〉 have been rescaled such that the near coincidence of the plotssignals the expected O(h2) convergence to conservation.The second conserved quantity that we monitor is the real-space particle flux,Jµ, given byJµ(t, r) = gµν∫∫pνmfdVpj . (2.49)In spherical symmetry, the only nonzero components of Jµ areJt = − απar2∫∫f(t, r, pr)dpr , (2.50)Jr =πar2∫∫prp¯tf(t, r, pr)dpr . (2.51)The divergence of the flux must remain zero as the system evolves—written explicitlywe have∇µJµ = 1α3a3r(− a3rJ˙tα+ a3rJtα˙+ arJrα2α′+ α3rJ ′ra − αrJta2a˙− α3rJrα′ + 2Jrα3a)= 0 . (2.52)Plots of the rescaled ℓ2 spatial norm of ∇µJµ as a function of time are shown inFig. 2.2(d)—again O(h2) convergence is observed.Independent Residual TestAs noted in Sec. 2.2.1, the θθ component of Einstein’s equation is not used in ourevolution scheme but must be satisfied in the continuum limit if our numerical resultsare valid. We thus define the residualEθθ ≡ Gθθ − 8πT θθ , (2.53)532.4. Numerical TechniquesFigure 2.2: Results of various diagnostic tests used to test the numerical solver. Theinitial data and mesh resolutions used here are typical of any of the 2D calculationsdescribed in the paper. A standard convergence testing methodology, using threecalculations with fixed initial data and mesh spacings h, h/2 and h/4, is employed.The coarsest mesh has nx × ny = nr × np = 128 × 128 grid points. Plots (a), (c)and (d) all display quantities that are residual in nature, i.e. which should tend tozero quadratically in the mesh spacing. Values from the h/2 and h/4 computationshave been rescaled by factors of 4 and 16, respectively, and the near-coincidence ofthe rescaled values thus demonstrates that all three quantities are converging at theexpected O(h2) rate. (a) Convergence of the l2 norm of the independent residual,‖Eθθ‖2, defined by (2.53). (b) Convergence factors (3.68) of the primary dynamicalunknowns. Here, convergence of the metric functions, α and a, is clearly secondorder, while that for the distribution function is better than O(h) but is not O(h2).This latter behaviour is to be expected since the finite volume method used to updatef is only first order in the vicinity of local extrema. (c) Convergence of the deviationin computed total mass, calculated from (2.45) and (2.46). (d) Convergence of theparticle flux divergence (2.52).542.4. Numerical TechniqueswhereGθθ = Gφφ= − 1rα3a3(−α2a∂α∂r+ α3∂a∂r+ α2r∂α∂r∂a∂r− α2ar∂2α∂r2+ a2αr∂2a∂t2− a2r∂α∂t∂a∂t), (2.54)and T θθ is given by (2.22). Then, using second-order finite differences to approx-imate all derivatives, we monitor the ℓ2 norm of Eθθ during the calculations. Weexpect ‖Eθθ‖2 to be O(h2) and Fig 2.2(a) shows that this is the case.Full-solution Convergence TestThe final check we perform is a basic convergence test of the primary dynamicalvariables, α, a and f . Denoting the values computed at resolution h for any ofthese by qh(t,X)—where X = r for α and a, and X = (r, pr) for f—we calculateconvergence factors, C(t; q), defined byC(t; q) =||qh(t,X)− qh/2(t,X)||l2||qh/2(t,X) − qh/4(t,X)||l2. (2.55)If our scheme is O(h2) convergent then it is easy to argue that C(t; q) should ap-proach 4 in the continuum limit. Plots of C(t; a), C(t, α) and C(t; f) are shown inFig. 2.2(b). Second order convergence of the geometric variables is apparent, whilethe behaviour of C(t; f) reflects the fact that the finite volume method we use isonly first-order accurate in the vicinity of extrema of f . Interestingly, at least atthe resolutions used here, the deterioration of the convergence of f does not appearto significantly impact that of the geometric quantities.2828This can be traced back to the fact that the geometric quantities are only related to f via theintegral of the density function over the momentum direction in the phase space. Finite volumemethods lose their point-wise second order accuracy in the vicinity of extrema points, but they arewell preserving with respect to the quantities derived by integrating over the finite volume cellssuch as the energy momentum tensor.552.5. ResultsFamily D f(0, r, pr, l) pG1 2 δ(l − l0)G(A, rc, pc) pcG2 2 δ(l − l0)G(A, rc, pc) l0G3 2 δ(l − l0)G(A, rc, 0) AG4 2 δ(l − l0) (G(A, rc, pc) + G(A, rc +∆r, pc +∆p)) pcG5 2 δ(l − l0)E(A,rc, pc) pcG6 2 δ(l − l0)E(A,rc, 0) AG7 2 δ(l − l1)G(A, r1, p1) + δ(l − l2)G(A, r2, p2) p1G8 3 exp(−(l − l0)2/∆l2)G(A, rc, pc) pcG9 3 exp(−(l − l0)2/∆l2)G(A, rc, 0) AG10 3 Θ(l − 5)Θ(15− l)E(A, rc, 0) ATable 2.1: Families of generic initial data used in the studies described in text. Thecolumns enumerate: (1) the label for the family, (2) the number, D, of phase-spacedimensions on which the distribution function depends (and therefore whether the2D or 3D code was used to generate the results), (3) the form of the initial data,f(0, r, pr, l) (see (2.57) and (2.58) for the definitions of G and E), and (4) the con-trol parameter, p, that was varied to study the critical behaviour. The quantitiesl0, l1, l2, rc, r1, r2, pc, p1, p2,∆r and ∆p that appear in the various specifications off(0, r, pr, l) are all parameters; i.e they have fixed scalar values in any given compu-tation.2.5 ResultsIn this section we describe the main results from our investigation of critical be-haviour in the Einstein-Vlasov model. We have used many different families ofinitial data in our studies and what we report below is based on a representativesample of those. As mentioned in the introduction, the numerical experiments fallinto three broad classes. The first uses massless particles and initial data whichhave some relatively arbitrary form in phase space. The second also uses mass-less particles but with initial conditions that represent perturbed static solutions.Finally, the third set is the same as the first but with massive particles. We willrefer to these classes as generic massless, near-static massless, and generic massive,respectively. In addition, the calculations can be categorized according to whetherl is a single fixed value, l0, (2D) or if the distribution function has non-trivial l-dependence (3D). The functional form of the various families considered, along withthe dimensionality of the corresponding PDEs and the parameter used for tuning562.5. Resultsto criticality are summarized in Table Generic Massless CaseHere we use initial distribution functions, f0 ≡ f(0, r, pr, l), that describe configura-tions of particles localized in r, pr and l, and that include various parameters whichcan be tuned to generate families of solutions that span the black hole threshold.Specifically, we setf(0, r, pr, l2) = S(r, pr)F (l) , (2.56)where S(r, pr) is given by either a gaussian function,G(r, pr ; A, rc, pc) ≡ A exp(−(r − rc)2∆2r− (pr − pc)2∆2p), (2.57)or the truncated bi-quadratic formE(r, pr; A, rc, pc) ≡Ar¯(1− r¯)p¯(1− p¯) 0 < r¯ < 1 ,0 < p¯ < 1 ,0 elsewhere,(2.58)where r¯ = (r−rc+∆r)/2∆r and p¯ = (pr−pc+∆p)/2∆p. Note that the dependenceof G and E on r and pr is suppressed in the abbreviated notation used in Table 2.1.For the 3D calculations, we use two types of angular momentum distribution: thefirst is a gaussian,F (l) = exp(−(l − l0)2∆l2), (2.59)while the second is uniform in l with cutoffs at some prescribed minimum andmaximum values, lmin and lmax, respectively,F (l) = Θ(l − lmin)Θ(lmax − l) . (2.60)It is important to point out that since the massless Einstein-Vlasov system is572.5. ResultsrT = 0  pr2 4 6 8 10−50512345x 10−3rT = 40  pr2 4 6 8 10−505012345x 10−3rprT = 120  2 4 6 8 10−50501234x 10−3r˜prT=200  1.5 2 2.5−4−202401234x 10−3Figure 2.3: Snapshots of the distribution function from a typical near-critical cal-culation, with evolution proceeding left to right, top to bottom (note the reductionin the range of radial coordinate in the last frame). The displayed results are fromfamily G8 (see Table 2.1) where pc—which is loosely the average momentum of theinitially imploding shell of particles—is the control parameter. As with all of thecalculations discussed in the results section, the control parameter has been tunedto roughly machine precision. In the early stages of the evolution we observe phasespace mixing and the ejection of some particles (the latter particularly visible asthe “tail” in the second frame). At intermediate times the system approaches astatic state which persists for a period that is long compared to the infall/dispersaltimescale characterizing weak field dynamics. We note that this is a 3D calculation,with f non-trivial in the l direction: for visualization purposes we have integratedover l to produce a quantity depending only on r and pr. Additionally, the first threeframes are plotted using the computational coordinate, r, while for the purposes ofdirect comparison with Fig. 2.4, the fourth uses the rescaled coordinate, r˜, definedby (2.64). We emphasize that at criticality f retains non-trivial dependence on pr;that is, although the geometry is static, the particle behaviour is still dynamic.582.5. ResultsrprT = 0  2 4 6 8 10−4−202424681012x 10−3rprT = 40  2 4 6 8 10−4−202424681012x 10−3rprT = 160  2 4 6 8 10−4−202424681012x 10−3r˜prT = 350  1.5 2 2.5−4−202400.0020.0040.0060.0080.010.012Figure 2.4: Snapshots of the distribution function for a near-critical calculation us-ing family G10. Here the tuning parameter is the overall amplitude, A, of the initialparticle distribution. As in the previous figure the sequence shows an approach to astatic state, but it is evident that the form of the distribution function at criticalityis significantly different in the two calculations. Due to the use of the rescaled radialcoordinate, r˜, the fourth frames of the two figures can be meaningfully compared.592.5. Resultsscale-free it has an additional symmetry relative to the massive case. Specifically,the equations of motion are invariant under the transformationt→ kt , (2.61)r → kr , (2.62)where k is an arbitrary positive constant. In order to meaningfully compare resultsfrom different initial data choices we must therefore adopt unitless coordinates inour analysis. We do this by rescaling t and r by the total mass, M⋆, of the puta-tively static solution which arises at criticality for any of the families that we haveconsidered (that is, M⋆ includes only the mass associated with that portion of theoverall matter distribution which appears to be static at criticality). Moreover, itis more natural and convenient to use central proper time, τ , rather than t itselfin the analysis. Thus, the results below are described using rescaled coordinates, τ˜and r˜, defined byτ˜ =τM⋆, (2.63)r˜ =rM⋆. (2.64)We note that under the scaling (2.61)–(2.62) the angular momentum transforms asl→ kl . (2.65)The process we use to generate near-critical solutions is completely standard forthis type of work. All of the family definitions described above and summarizedin Table 2.1 contain multiple parameters that can be used to tune to the blackhole threshold and, consistent with what has been found in many other previousstudies of black hole critical phenomena, we find that which particular parameter isactually varied is essentially irrelevant for the results. Having chosen some specific602.5. Resultsparameter, p, to vary, any critical search begins by determining an initial bracketinginterval, [pl, ph], in parameter space such that evolutions with pl and ph lead todispersal and black hole formation, respectively. We then narrow the bracketinginterval using a bisection search on p, predicating the update of pl or ph on whetheror not a black hole forms. The search is continued until (ph − pl)/ph ∼ 10−15, sothat p⋆ is computed to about machine precision (8-byte floating point arithmetic).The value of pl at the end of this process corresponds to what we dub the marginallysub-critical solution.Quite generically, as we tune any family to a critical value p⋆, the phase spacedistribution function appears to settle down to a static solution which, as p →p⋆, persists for a time that is long compared to the characteristic timescale forimplosion and subsequent dispersal of the particles in the weakly-gravitating limit.Representative illustrations of this behaviour are shown for marginally sub-criticalevolutions from two distinct initial data families in Fig. 2.3 (family G8 in Table 2.1)and Fig. 2.4 (family G10). Similarly, the spacetime geometry–encapsulated in themetric functions a and α—also becomes increasingly time-independent as criticalityis approached. Fig. 2.5 displays the evolution of the ℓ2-norm of the time derivativeof a during marginally sub-critical evolution for family G1. We thus have strongevidence that the critical solutions that we are finding are static—characteristic oftype I critical behaviour—and consistent with what has been observed previouslyfor the case of the massive Einstein-Vlasov system.Further evidence for generic type I transitions in the model is provided by ob-servations of lifetime scaling of the form (2.1) near criticality, which is expected ifthe critical solutions are one-mode unstable. Typical results from calculations usingfamilies G1, G4, G8 and G10 are shown in Fig. 2.6: the linearity of the lifetimeof the static critical configuration as a function of ln |p − p⋆| is apparent. We haveobserved such scaling for all of the families that we have studied (in both the 2D and3D cases) and Table 2.2 provides a summary of the measured values of the scaling612.5. ResultsFigure 2.5: Time evolution of ‖∂ta(t, r)‖2 from a marginally sub-critical calcula-tion using family G1. The plot provides strong evidence that the geometry of thethreshold solution is static, a characteristic feature of type I behaviour.622.5. ResultsFigure 2.6: Lifetime scaling of near-critical configurations for families G8, G1, G10and G4 (top to bottom and noting that G10 and G8 are 3D calculations whilethe others are 2D). Here the symbols plot estimates of the amount of time thestate of the system is well approximated by the static critical solution—measuredin units of the rescaled proper time defined by (2.63)—as a function of ln |p − p⋆|.The lines are least squares fits to τ = −σ ln |p − p⋆| where σ is the reciprocal ofthe eigenvalue (Lyapunov exponent) corresponding to the presumed single growingmode of the critical solution. To the estimated level of accuracy in our calculationsthe measured values of σ are the same for the four families. However, we cannotstate with certainty that there is precise universality in this regard.632.5. ResultsFigure 2.7: Radial metric function a(r˜) at criticality for families G8, G1, G10 andG4. The results plotted here, together with those displayed in Fig. 2.8, show thatthere is relatively little variation in the geometry of the static critical configurationas a function of the specifics of the initial data. The inset plots the deviation in afor families G1, G10 and G4 relative to G8.642.5. ResultsFigure 2.8: Lapse function α(r˜) at criticality for families G8, G1, G10 and G4. Thecomments made in the caption of the previous figure apply here as well.652.5. Resultsexponent, σ.We note that the specific form of the matter configuration at criticality exhibitssignificant dependence on the family of initial data that is used to generate thecritical solution. This can be seen, for example, by comparing the last frames ofFigs. 2.3 and 2.4. On the other hand, as illustrated in Fig. 2.7 and Fig. 2.8, thegeometry of the critical state is relatively insensitive to the initial conditions.The spacetime geometry can be characterized by the central red shift, Zc definedby (2.44), and the unitless compactness parameter, Γ, defined byΓ = maxr2mr. (2.66)For the families considered in this section the values of Γ and Zc fall in the ranges0.79 . Γ . 0.81 , (2.67)2.4 . Zc . 2.5 . (2.68)As discussed in the next section, these ranges are relatively small in comparison tothose found in our investigation of critical behaviour using nearly-static initial data.What is striking about the results assembled in Table 2.2 is that there appearsto be a small variation, at most, in the time scaling exponent associated with thecritical solutions produced from our generic initial conditions. Specifically, the datais consistent withσ = 1.4 ± 0.1 , (2.69)and we emphasize that this concordance arises despite the significant observed vari-ation in the phase-space distribution of the particles among the various criticalsolutions.662.5. ResultsFamily l0 σ Family l0 σG1 5 1.32 ± 0.08 G3 12 1.36 ± 0.06G1 6 1.35 ± 0.07 G4 12 1.37 ± 0.05G1 7 1.36 ± 0.06 G5 12 1.44 ± 0.06G1 8 1.33 ± 0.06 G6 12 1.43 ± 0.04G1 9 1.33 ± 0.06 G7 6 & 12 1.37 ± 0.07G1 10 1.32 ± 0.06 G8 10 1.35 ± 0.05G1 11 1.35 ± 0.05 G9 10 1.36 ± 0.05G1 12 1.37 ± 0.05 G10 - 1.40 ± 0.05G2 - 1.36 ± 0.07Table 2.2: Summary of measured lifetime scaling exponents for the masslessEinstein-Vlasov model from experiments using the various initial data families enu-merated in Table 2.1. In addition to the overall functional form of the initial dis-tribution functions, a key parameter that varies among the sets of calculations isl0, which is the angular momentum of any and all particles for families G1, G2–G6(2D) and the center of the angular momentum distribution for families G8 and G9(3D). (l0 is the tuning parameter for G6, and family G7 is another special casewhere the initial data is comprised of a superposition of two shells of particles, eachhaving a distinct angular momentum parameter. Since angular momentum is a con-served quantity there is no mixing of the two distributions during the evolution.)For simplicity of presentation we have not listed the other parameters defining thedifferent initial configurations. Quoted uncertainties in the values of σ are basedon variations in the total mass of the system during the evolutions and comparisonwith results computed at lower resolution. Typical grid sizes used for the listedresults are nr × np = 1024 × 1024 (2D) or nr × np × nl = 256 × 128 × 64 (3D). Tothe level of accuracy in our calculations we find consistency with a single value ofthe scaling exponent, σ = 1.4 ± 0.1.672.5. Results2.5.2 Near-static Massless CaseOur second approach to study critical solutions in the massless Einstein-Vlasov sys-tem starts with the construction of static initial data using the procedure describedin Sec. 2.3. We specialize the general form (2.41) toΦ(E, l) = C(1− E/E0)bΘ(E0 − E)δ(l − l0) , (2.70)where E0 is a given cutoff energy and C, b and l0 are additional adjustable pa-rameters. Here we focus exclusively on the case of fixed angular momentum (2Dcalculations) since the results of the previous section suggest that the essential fea-tures of the critical solutions are not significantly dependent on whether or not fhas non-trivial dependence on l. In addition, from the scale free symmetry in thesystem (see (2.61) and (2.65)), we can conclude that varying the value of angu-lar momentum is equivalent to rescaling the radial coordinate. Therefore, withoutloss of generality we can set l to an arbitrary fixed value, eliminating one of theparameter-space dimensions in our surveys. Additionally, so that we can meaning-fully compare results from different initial conditions, we again rescale the radialcoordinate by the total mass of the system (2.64). Furthermore, by virtue of thetransformation (2.43), the static profiles depend on E0 only through the ratio E0/α0and, since it simplifies the numerical analysis, we actually use this ratio as one ofthe control parameters.For specified values of the free parameters C, b and E0/α0, we integrate equa-tions (2.29)–(2.32) outward until we reach a radial location, rX , where the particledensity Φ(E, l) vanishes. We then extend the solution for a and α to the outerboundary of the computational domain by attaching a Schwarzschild geometry withthe appropriate mass.We note that not all choices of the three free parameters lead to distributionfunctions with compact support—that is, with f(0, r, pr) ≡ 0 for r greater than682.5. Resultssome rX—so that the configuration represents a single shell of particles. Indeed, byexamining the expression for the particle energy in the massless case:E(r, pr) = α(r)√(pr/a)2 + (l/r)2 , (2.71)we see that, for pr sufficiently small, E(r, pr) can remain below the cutoff E0 forlarge r. In practice this will yield solutions with multiple shells, where Φ vanishesat rX , but then becomes non-zero on a infinite number of intervals in r (in generalthese intervals can be disjoint or contiguous, as has previously been seen in [43]).Although it might be interesting to consider the critical dynamics of multiple-shellsolutions, we do not do so here. We also note that for given values of b and E0/α0we find solutions with a distinct shell (i.e. where Φ does vanish at some radius) onlyfor a certain range of C, but that range can span several orders of magnitude.Fig. 2.9 shows the distribution function for four sample static configurationsconstructed as described above, with the associated geometrical variables plotted inFig. 2.10. Relative to the apparently static solutions generated by tuning genericinitial data, the family-dependence of both the distribution function and metricvariables here is much more pronounced.One interesting way of characterizing the static solutions is to plot the compact-ness parameter, Γ, defined by (2.66), as a function of the central redshift, Zc. Wedo this for a large number of configurations in Fig. 2.11 where, as described in moredetail in the caption, each set of points results from a two-dimensional parameterspace survey wherein both E0/α0 and C are varied. The fact that the solutions fromeach of these surveys tend to “collapse” to one-dimensional curves in Zc–Γ space isstriking and we do not have any argument at this time for why this should be so.All of the static solutions that we have found satisfy Buchdahl’s inequality,Γ < 8/9, originally derived in the context of fluid matter [59], and the most compactconfigurations are quite close to that limit. Here it is crucial to note that Andre´assonhas proven rigorously that the Buchdahl inequality is satisfied by any static solution692.5. Resultsr˜Prb = 1 Zc = 2.32 C = 0.1  1.8 2 2.2 2.4−3−2−1012300.0020.0040.0060.0080.010.012r˜Prb = 2 Zc = 2.22 C = 0.1  1.6 1.8 2 2.2 2.4 2.6−5050.511.522.533.5x 10−3r˜Prb = 2 Zc = 2.04 C = 103  2.16 2.18 2.2 2.22 2.24 2.26−0.500.500.˜Prb = 4 Zc = 2.17 C = 10  1.9 2 2.1 2.2 2.3 2.4−4−20240246810x 10−3Figure 2.9: Sample static phase space configurations computed from theansatz (2.70) using different choices of adjustable parameters. Note that althoughwe use the rescaled radial coordinate r˜ in all of the plots, the ranges in r˜, pr and fvary from frame to frame. Clearly, there is a strong dependence of f on the chosenparameter values. As described in more detail in the text, for any given values ofb and Zc there is a finite range of C for which we find static solutions where f hascompact support.702.5. ResultsFigure 2.10: Plots of the radial metric function, a(r), and differential particlenumber, dN(r)/dr, for the configurations shown in Fig. 2.9. The graphs of dN(r)/drhighlight the fact that the critical solutions are shell-like, with a thicknesses andeffective densities that are strongly dependent on the choice of parameters in (2.70).of the spherically symmetric Einstein-Vlasov system [58]. Further, he has demon-strated that one can construct static shell-like configurations which, in the limit ofinfinitesimal thickness in r, can have Γ arbitrarily close to 8/9. Although not explic-itly mentioned in [58], it is clear that his proof is valid for m = 0. Given the natureof Andre´asson‘s result, the observation that our solutions satisfy the bound clearlyamounts to little more than additional evidence that our calculations are faithful tothe model under study. However it is interesting that the highest values of Γ seen inFig. 2.11—and which plausibly are approaching 8/9—are associated with very thinshell-like solutions. Additionally, for the configurations we have studied (not all ofwhich are represented in Fig. 2.11) there is apparently also a lower bound on thecompactness, Γ ∼ 0.81. Finally, the ranges of Γ and Zc spanned by the explicitlystatic solutions0.80 . Γ . 0.89 , (2.72)712.5. ResultsFigure 2.11: The value of Γ = maxr(2m/r) versus central redshift, Zc, for variousstatic solutions. Each set of points comprises several thousand distinct solutionsand comes from a two-dimensional parameter space survey, in which both C andE0/α0 are varied. Although for given b and E0/α0 we can only find acceptable staticsolutions in certain ranges of C, those ranges can span several orders of magnitude.However, for fixed b the solutions tend to collapse to near-linear loci in Zc–Γ space,and the inset graph, which plots the deviation of the data from a linear least squaresfit, is intended to emphasize this behaviour. More detailed examination of thedata suggests that the configurations do not lie precisely along one-dimensionalcurves, but additional study would be required to determine whether this is reallythe case. The solutions apparently satisfy the Buchdahl inequality Γ < 8/9 (alsoseen in the calculations reported in [43] for the massive case), as is expected fromAndre´asson‘s rigorous results [58]. Moreover, there also seems to be a lower boundon the compactness, Γ ∼ 0.81.722.5. Resultsb Zc C δf σ1 2.32 0.1 δf1 1.45 ± 0.051 2.23 0.3 δf1 1.45 ± 0.042 2.22 0.1 δf1 1.43 ± 0.044 2.17 10 δf1 1.43 ± 0.042 2.35 0.1 δf1 1.40 ± 0.052 2.35 0.1 δf2 1.40 ± 0.052 2.35 0.1 δf3 1.40 ± 0.05Table 2.3: Measured lifetime scaling exponent for explicitly static solutions con-structed from ansatz (2.70) with various choices of the adjustable parameters b,E0/α0 and C (Zc is effectively controlled by E0/α0, but is determined a poste-riori), and the different types of perturbations, δf , enumerated in (2.75)–(2.77).Proceeding from the assumption that the static solutions are characterized by asingle unstable mode, we anticipate that the computed value of σ associated with aspecific configuration (i.e. for given b, Zc and C) should be independent of the formof δf , and this is precisely what we observe (compare rows 1 and 2, and 5, 6 and 7).However, we also see once again that there is little, if any, variation in the scalingexponent with respect to the underlying critical solution: the results in the tableare consistent with σ = 1.43 ± 0.072.0 . Zc . 2.4 , (2.73)are larger than those seen for the tuned generic data, consistent with the commentabove concerning the relatively large variations in the metric variables as well as thedistribution function.Using our evolution code, we investigate the relation of the explicitly-static so-lutions to critical behaviour in the model as follows. For initial conditions we setf(0, r, pr, l2) = f0(r, pr, l2) + (A− 1)δf(r, pr , l2) , (2.74)where f0 is a static configuration, δf(r, pr, l2) is some given perturbation functionwith at least roughly the same support as f0, and A is a tunable parameter whichcontrols the amplitude of the perturbation. Clearly, A = 1 results in initialization732.5. Resultswith the static solution itself. We have experimented with the following three choicesfor the perturbation function:δf1(r, pr, l2) = f0(r, pr, l2) , (2.75)δf2(r, pr, l2) = sin(2πf0(r, pr, l2)fmax), (2.76)δf3(r, pr, l2) = f0(r, pr, l2)(fmax − f0(r, pr, l2))pr , (2.77)where fmax is the maximum of f0 over the computational domain. We then performstandard tuning experiments in which we vary A to isolate a threshold solution.Interestingly, we find strong evidence that all of the static solutions based on (2.70)that we have found sit at the threshold of black hole formation, so that setting A > 1results in black hole formation while taking A < 1 results in complete dispersal ofthe matter (or vice versa, dependent on the precise form of δf). As should be sus-pected then, and as is shown for four families in Fig. 2.12, the solutions generated bydynamically evolving the perturbed static configurations exhibit time scaling—thisstrongly suggests that the time-independent solutions are all one-mode unstable.Table 2.3 provides a summary of the time-scaling exponents we have measured fora set of experiments based on four distinct static solutions and the three differenttypes of perturbation defined by (2.75)–(2.77).As was the case for the generic families, the measurements here indicate thatalthough the static solutions display significant variation in both the distributionfunction and geometric variables, there is little variation in the scaling exponent.Here we findσ = 1.43 ± 0.07 . (2.78)Recalling (2.69), and given the estimated uncertainty in our calculations, we can not742.5. ResultsFigure 2.12: Lifetime scaling computed from families of initial data based on thestatic configurations plotted in Figs. 2.9 and 2.10. The tuning parameter in thisinstance controls the amplitude of a perturbation that is added to the base solution(here we used the form δf1 (2.75)) and, in all cases, the sign of the perturbationdetermines whether the evolution leads to dispersal or black hole formation. Theresults shown here provide evidence that the static configurations calculated from theansatz (2.70) act as type I critical solutions. Additionally, we see that there is verylittle variation in the measured scaling exponents, σ, which are again determinedvia least squares fits to (2.1).752.5. Resultsexclude the possibility that σ is truly universal for the massless-sector critical solu-tions which we have constructed. Particularly given the variation in the spacetimegeometries involved, constancy of the eigenvalue of the unstable mode associatedwith criticality would be truly remarkable. However, even if σ does span some finiterange, the apparent tightness of that range is an aspect of critical behaviour in themassless system that begs understanding.Finally, we note that the static critical solutions from the generic calculations arecharacterized by compactness, Γ ∼ 0.8, which is at the low end of the range spannedby the explicitly static solutions. We do not yet know whether a more extensiveparameter space survey of generic data could produce critical configurations withlarger Γ, and it would be interesting to further investigate this issue.2.5.3 Generic Massive CaseFollowing previous studies [34–36], we have also examined the case where the par-ticles have rest mass and find results that are in general agreement with the earlierwork, including strong evidence for the existence of static solutions at the black holethreshold that exhibit lifetime scaling. However, we note that in both [35] and [36]the initial data configurations were kinetic energy dominated. For example, a typ-ical calculation in [35] used unit particle mass and f(0, r, pr, l) which was gaussianin the three coordinates with characteristic values r ∼ 3, pr ∼ 1 and l ∼ 3. Fromexpression (2.35) for the particle energy we can thus infer that the initial data setshad kinetic energy about an order of magnitude larger than rest mass energy. Thuswe expect that those previous results should be similar to what we see for masslessparticles. Indeed, taking into account the different time parameterization used (tnormalized to coincide with property time at infinity), the scaling exponents quotedin [35] are consistent with our results.Table 2.4 lists the values of the time scaling exponent we have determined in themassive case for the various types of initial data defined in Table 2.1. We note that762.5. Resultsthe initial data families that are used include ones that are very similar to thoseadopted in [35] and [36]. We see that the time scaling exponents are in fact closeto those measured in the massless calculations, although the spread in the values isnoticeably larger here (as it was in [35] and [36]). This increased spread is almostcertainly due to the particle mass—i.e. the evolutions are not completely kineticenergy dominated.Paralleling what was done in Sec.2.5.2, as well as in [43], we can use perturba-tions of our explicitly static solutions in the massive sector to investigate criticalbehaviour. Here there is a larger function space of static configurations, especiallysince we can construct solutions with positive binding energy, Eb, defined byEb ≡M0 −M , (2.79)where M0 is the total rest mass and M is the ADM mass. Moreover, we can buildparameterized sequences of solutions that transition between positive and negativeEb, completely analogously to what can be done for perfect fluid models of generalrelativistic stars. As in the perfect fluid case, we anticipate that: 1) solutionswith Eb > 0 will be perturbatively stable, 2) there will be a change of stability atEb = 0, and 3) for at least some range of Eb < 0, the static configurations will beone-mode unstable, and thus should constitute type I critical solutions. We haveperformed additional calculations that confirm these expectations. In particular, wewere able to build a static solution with Eb negative, but relatively close to 0, whichdid lie at the black hole threshold and which had an associated scaling exponentσ = 3.0±0.1. This value of σ is clearly distinct from those listed in Table 2.4. Thus,in contrast to the massless case where we can not conclusively state anything aboutpossible variations in σ for type I critical solutions, we are confident that σ is is notuniversal in the massive case. In fact, were we able to construct static configurationswith Eb → 0−, we assume that we would find σ → ∞. Again, these observationsand conjectures are entirely consistent with previous studies of the Einstein-Vlasov772.6. Summary and DiscussionFamily l0 Zc σ Family l0 Zc σG1 5 2.47 1.32± 0.14 G1 12 2.28 1.46 ± 0.07G1 6 2.39 1.47± 0.13 G2 - 2.39 1.44 ± 0.09G1 7 2.31 1.44± 0.08 G3 9 2.29 1.54 ± 0.07G1 8 2.37 1.49± 0.08 G4 9 2.43 1.49 ± 0.08G1 9 2.41 1.49± 0.08 G8 10 2.24 1.38 ± 0.14G1 10 2.34 1.48± 0.07 G9 10 2.41 1.59 ± 0.15G1 11 2.23 1.54± 0.07Table 2.4: Summary of measured lifetime scaling exponents for the massive Einstein-Vlasov model from experiments using the various initial data families enumeratedin Table 2.1. The results quoted here derive from calculations that parallel thosedescribed in Table 2.2 for the massless system. In contrast to the massless case, theobserved variation in σ is significant.system, as well as work with gravitationally compact stars modelled with perfectfluids or bosonic matter.2.6 Summary and DiscussionWe have constructed a new numerical code to evolve the Einstein-Vlasov system inspherical symmetry using an algorithm where the distribution function f(t, r, pr, l2)is directly integrated using finite volume methods. This approach eliminates thestatistical uncertainty inherent in the particle-based techniques that have been usedin previous studies. To reduce computational demands at a given discretization or,more importantly, to allow for higher resolution, we can also run the code in a 2Dmode where l2 is some fixed scalar constant so that f depends on only r and pr.We have used the code to perform extensive and detailed surveys of the criticalbehaviour in the model with a particular focus on the case where the particlesare massless. We note that we are unaware of any previous dynamical numericalcalculations pertaining to the massless sector.Our results derive from two classes of initial configurations. In the first theinitial states represent imploding shells of particles well removed from the origin,while the second involves perturbations of configurations that are precisely static782.6. Summary and Discussionby construction. Although time-independent solutions of the massive system havebeen constructed and analyzed previously, to our knowledge the static states wehave found in the massless sector are the first of their kind. Within each class wehave studied numerous specific forms for the initial data and, for the near-static cal-culations, the perturbations that are applied to generate the threshold behaviour.In all cases we find strong evidence for a Type I critical transition including: 1) afinite black hole mass at threshold and 2) lifetime scaling of the form (2.1). The ob-servations are all consistent with the standard picture for Type I behaviour, namelya static critical solution with one unstable perturbative mode. Here we emphasizethat—as is the case for any numerical study of critical behaviour—it is very diffi-cult to preclude the existence of additional unstable modes. However, the degree towhich the scaling laws are satisfied suggests that if such modes do exist they havegrowth rates significantly smaller than the dominant one.For generic initial data with massless particles, we have found that there is aconsiderable variation in the morphology of f among the different critical solutionswe have computed and, to a lesser extent, in the details of the spacetime geometriesencoded in a(t, r) and α(t, r). Interestingly though, there is relatively little variationin the time scaling exponents that we have measured: all seem to be in the range1.3 . σ . 1.5.In the case of near-static initial conditions with m = 0 the key results are quitesimilar. Again, there is a large variation in the functional form of the distributionfunction at threshold. In this instance this can be seen as a direct reflection ofthe freedom inherent in the ansatz (2.70) which involves the specification of twoessentially arbitrary functions. Not surprisingly, there is thus a more noticeablerange in the geometries at criticality relative to the generic calculations, as can beclearly seen, for example, through examination of quantities such as the compactnessand central redshift. Once again, however, we observe only a small dispersion inthe measured scaling exponents. Specifically, across all near-static families that we792.6. Summary and Discussionhave examined we find σ = 1.43 ± 0.07.Thus, considering all of the calculations that we have performed, we have indi-cations of at least a weak form of universality of the time-scaling exponent in themassless Einstein-Vlasov model. Here we note that as mentioned in the introduc-tion, the calculations reported in [35] were also suggestive of a universal value ofσ and perhaps of the critical geometry. Those computations used a non-zero massand, as also discussed previously, the work of [37, 43] established that the spacetimestructure at criticality could not be universal in the massive model. However, asnoted in Sec. 2.5.3 the initial data families used in [35] were kinetic energy domi-nated (effectively massless), and so there is no contradiction between what was seenthere (and here) and [37, 43].In all of our calculations, and in accord with Andre´asson’s proof of the Buchdahlinequality in the model [58], we observe that the gravitational compactness satisfiesΓ < 8/9 , with thin shell-like solutions coming closest to saturating the bound.We also want to emphasize an additional feature of the massless model thatis apparent from our calculations: the particle angular momentum does not havea significant impact on the features of the critical solution (apart from the obvi-ous fact that the particles do have angular momentum in all of our computations).Heuristically, this can be at least partly ascribed to the scaling symmetry (2.61)–(2.62). The symmetry effectively reduces the number of free parameters—relative toa naive analysis—available for variation in the search for critical solutions. Specifi-cally, given any distribution of the form f(r, pr)δ(l−l1), where l1 is fixed, we can mapto a distribution f ′(r, pr)δ(l − l2), with l1 6= l2, which has an associated geometrythat is diffeomorphic to the original.Given that there is clearly no universality of the fundamental dynamical variablesat threshold, the fact that the variation in σ is, at most, small is a feature of thecalculations for which we currently have no explanation. Additionally, as discussedin the introduction, the argument advanced in [52] suggests that there should be no802.6. Summary and Discussiontype I behaviour in the Einstein-Vlasov system for either the massless or massivemodels. At this time, we do not understand how—if at all—this argument can bereconciled with our current results and those from previous numerical studies.A direct analysis of the perturbations of the critical solutions—especially theprecisely static ones—would be very helpful at this point. Starting with the perfect-fluid work of Koike et al [25], perturbation analyses of the critical configurations inmany different models have been extremely effective in advancing our understandingof black-hole critical phenomena. In particular, relative to measurements madethrough direct solution of PDEs and tuning experiments, perturbative methodscan provide highly accurate values for the eigenvalues of the unstable modes (or,equivalently, for the scaling exponents). However, in our case the task of explicitlyconstructing perturbations is significantly complicated by the fact that there is noone-to-one correspondence between the geometry and the phase-space distributionof the particles. So far we have been unable to formulate a well-defined approach tocomputation of the perturbations and will have to leave that for future work.Finally, it would be interesting to extend this work to the Einstein-Boltzmannsystem, where the introduction of explicit interactions between particles would pro-vide the means to investigate the connection between criticality in phase-space-basedmodels and hydrodynamical systems. This in turn might lead to a more fundamentalunderstanding of critical collapse in fluid models.81Chapter 3Black Hole Critical Behaviourwith the Generalized BSSNFormulation3.1 IntroductionIn this paper29 we investigate the application of the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of Einstein’s equations [31, 32], as well as the dy-namical coordinate choices typically associated with it, within the context of criticalgravitational collapse. The BSSN formulation is a recasting of the standard 3+1Arnowitt-Deser-Misner (ADM) [27] equations that is known to be strongly hyper-bolic [60, 61] and suitable for numerical studies. It has been widely used in numericalrelativity and provides a robust and stable evolution for the spacetime geometry.Most notably, various implementations of this formulation have allowed successfulcomputation of dynamical spacetimes describing binaries of gravitationally-compactobjects [3, 13, 14]. The standard gauge choices in BSSN—namely the 1+log slicingcondition [62] and the Gamma-driver shift condition [63]—are partial differentialequations (PDEs) of evolutionary type. Furthermore, the BSSN approach results ina set of so-called free evolution equations, meaning that the Hamiltonian and mo-mentum constraints are only solved at the initial time. Thus, once initial data has29This chapter is published in: Akbarian A. and Choptuik M. W. “Black hole critical behaviorwith the generalized BSSN formulation”. Phys. Rev. D92, 084037 (2015).823.1. Introductionbeen determined, one only has to solve time-dependent PDEs in order to computethe geometric variables in the BSSN scheme. In particular, during the evolutionthere is no need to solve any elliptic equations, which in general could arise eitherfrom the constraints or from coordinate conditions. This is advantageous since itcan be quite challenging to implement efficient numerical elliptic solvers.In addition to the BSSN approach, the numerical relativity community hasadopted the generalized harmonic (GH) [64] formulation of Einstein’s equations,which is also strongly hyperbolic and has performed very well in simulations ofcompact binaries [3, 12]. Like BSSN, the GH formulation is of evolutionary typeso that all of the metric components satisfy time-dependent PDEs. It too uses dy-namical coordinate choices: in this case one needs to provide a prescription for theevolution of the harmonic functions defined by Hµ ≡ xµ.Despite the tremendous success of these hyperbolic formulations in evolvingstrongly gravitating spacetimes containing black holes and neutron stars, they havenot seen widespread use in another area of strong gravity physics typically stud-ied via numerical relativity, namely critical phenomena in gravitational collapse.First reported in [4] and briefly reviewed below, critical phenomena emerge at thethreshold of black hole formation and present significant challenges for thorough andaccurate computational treatment. The original observation of critical behaviour aswell as many of the subsequent studies were restricted to spherical symmetry (fora review, see [5, 23]) and there is a clear need to extend the work to more genericcases. In this respect the BSSN and GH formulations would appear to be attrac-tive frameworks. However, it is not yet clear if these hyperbolic formulations, inconjunction with the standard dynamical gauge choices that have been developed,will allow the critical regime to be probed without the development of coordinatepathologies. Particularly notable in this regard is an implementation of the GHformulation that was employed by Sorkin and Choptuik [65] to study the criticalcollapse of a massless scalar field in spherical symmetry. Despite extensive exper-833.1. Introductionimentation with a variety of coordinate conditions, the code that was developedwas not able to calculate near-critical spacetimes: coordinate singularities invari-ably formed once the critical regime was approached. A natural question that thenarises is whether the BSSN formulation (including the standard dynamical gaugechoices used with it) is similarly problematic or if it provides an effective frameworkto study critical phenomena.Here we begin the task of addressing this question by revisiting the model ofspherically symmetric massless scalar collapse. We use a generalization of the BSSNformulation due to Brown [66] that is well suited for use with curvilinear coordinates.The choice of a massless scalar field as the matter source has the great advantagethat the nature of the critical solution is very well known [67–72], making it straight-forward for us to determine if and when our approach has been successful. We notethat although the calculations described below are restricted to spherical symmetryour ultimate goal is to develop an evolutionary scheme—including gauge choices—that can be applied to a variety of critical phenomena studies in axial symmetryand ultimately generic cases.We now briefly review the main concepts and features of black hole criticalphenomena that are most pertinent to the work in this paper. Full details andpointers to the extensive literature on the subject may be found in review articles[5, 23].Critical phenomena in gravitational collapse can be described as a phase transi-tion, analogous to that in a thermodynamical system. Under certain assumptions,a matter source coupled to the Einstein gravitational field will evolve to one of twodistinct final phases. On the one hand, weak initial data will eventually disperse toinfinity leaving flat spacetime as the end state. On the other hand, sufficiently strongdata will develop significant self gravitation and then collapse, resulting in a finalphase which contains a black hole. Quite generically, remarkable behaviour emergesat and near the transition between these phases, and this behaviour is precisely843.1. Introductionwhat we mean by the critical phenomena in the system under consideration.It transpires that there are two broad classes of critical phenomena that canbe distinguished by the behaviour of the black hole mass at threshold. The classof interest here, known as type II, is characterized by infinitesimal mass at thetransition. Further, the black hole mass, MBH , satisfies a scaling law:MBH ∼ |p− p⋆|γ , (3.1)where p is an arbitrary parameter that controls the strength of the matter sourceat the initial time, p⋆ is the parameter value at threshold and the mass scalingexponent, γ, is a constant that is independent of the choice of the initial data. TypeII behaviour is also characterized by the emergence of a unique solution at thresholdwhich is generically self-similar. In some cases, including the massless scalar field,the self-similarity is discrete. Specifically, in spherically symmetric critical collapsewith discrete self-similarity (DSS), as p→ p⋆ we findZ⋆(ρ+∆, τ +∆) ∼ Z⋆(ρ, τ) , (3.2)where Z⋆ represents some scale-invariant component (function) of the critical solu-tion. Here ρ ≡ ln(rS) and τ ≡ ln(TS − T ⋆S ) are logarithmically rescaled values ofthe areal radius, rS, and polar time, TS, respectively, and T⋆S is the accumulationtime at which the central singularity associated with the DSS solution forms. TShas been normalized so that it measures proper time at the origin. As with γ, theechoing (rescaling) exponent, ∆, is a universal constant for a specific matter source;i.e. it is independent of the form of the initial data.Another feature of type II collapse, intimately related to the self-similarity ofthe critical solution, is that the curvature can become arbitrarily large: in the limitof infinite fine-tuning, p→ p⋆, a naked singularity forms. Furthermore, the echoingbehaviour (3.2) results in the development of fine structure in the solution around the853.1. Introductioncenter of the scaling symmetry. Observing this structure and measuring the echoingexponent ∆ associated with it requires a code that can reliably evolve solutions veryclose to the critical spacetime and that provides sufficient numerical resolution inthe vicinity of the accumulation point (rS, TS) = (0, T⋆S ).As mentioned above, most studies of critical phenomena have assumed spheri-cal symmetry. This is particularly so for the case of type II behaviour where theresolution demands dictated by the self-similarity of the critical solutions makesmulti-dimensional work extremely computationally intensive. As far as we know,the only work in spherical symmetry to have used a purely evolutionary approachbased on the BSSN or GH forms of the Einstein equations is [65] which, as we havenoted, was not successful in isolating the critical solution.30 In axisymmetry therehave been two investigations of type II collapse of massless scalar fields [74, 75],and several of type II collapse of pure gravitational waves (vacuum) [76–80]. Ofthese, only Alcubierre et al.’s [77] and Sorkin’s [80] calculations of vacuum collapseadopted hyperbolic formalisms, and only the scalar field calculations—which em-ployed a modified ADM formulation and partially constrained evolution—were ableto completely resolve the critical behaviour, including the discrete self-similarity ofthe critical solutions. In the fully three-space dimension (3D) context there havealso been a few studies of type II collapse to date. Perhaps most notable is the recentwork of Healy and Laguna [81] which used a massless scalar field as a matter sourceand the BSSN formulation with standard dynamical gauge choices. The authorswere able to observe the mass scaling (3.1) with a measured γ ≈ 0.37 consistentwith calculations in spherical symmetry. However, they were not able to conclu-sively see the discrete self-similarity of the critical solution; in particular they couldnot accurately measure the echoing exponent, ∆. This shortcoming was attributedto a lack of computational resources rather than a breakdown of the underlyingmethodology, including the coordinate conditions that were adopted. Finally, there30However, see [73] for an investigation of type II behaviour in the collapse of a scalar field in2+1 AdS spacetime that employs an ad hoc free evolution scheme.863.1. Introductionhave been two attempts to probe the black hole threshold for the collapse of puregravitational waves in 3D [82, 83]. Both employed a BSSN approach with, for themost part, standard dynamical gauge choices. In both cases problems with thegauge apparently precluded calculation near the critical point (although resolutionlimitations may also have been an issue) and neither the mass scaling nor the echoingexponent could be be estimated in either study.We can thus summarize the state of the art in the use of hyperbolic formulationsfor the study of type II critical collapse as follows: to our knowledge there has beenno implementation of a fully evolutionary scheme, based on either BSSN or GH,that has allowed for evolution sufficiently close to a precisely critical solution toallow the unambiguous identification of discrete self-similarity (or continuous self-symmetry for that matter). Again, and particularly in light of the experience of [65],the key aim of this paper is to investigate the extent to which it is possible to usea BSSN scheme to fully resolve type II solutions. A major concern here is theappropriate choice of coordinate conditions, not least since dynamical gauge choicescan be prone to the development of gauge shocks and other types of coordinatesingularities [84, 85]. Such pathologies could, in principle, prevent a numericalsolver from evolving the spacetime in or near the critical regime.Now, as Garfinkle and Gundlach have discussed in detail [86], an ideal coordi-nate system for numerical studies of type II collapse is one which adapts itself tothe self-similarity: for the DSS case this means that the metric coefficients and rel-evant matter variables are exactly periodic in the coordinates in the fashion givenby (3.2). Clearly, if the coordinate system is adapted, then other than at the nakedsingularity—which is inaccessible via finite-precision calculations—it should remainnon-singular during a numerical evolution. One can then argue that ensuring thatthe numerical scheme has adequate resolution will be the key to successful simula-tion of the critical behaviour. At the same time, it is also clear that there will becoordinate systems which do not necessarily adapt but which nonetheless remain873.1. Introductionnon-singular during critical collapse, at least over some range of scales, and whichare therefore potentially useful for numerical calculations. We will see below thatthere is strong evidence that the coordinate systems we have used belong to thelatter class, and weaker evidence that they do adapt to the self-similarity.Another potential source of problems, which is not specific to hyperbolic for-mulations, relates to our restriction to spherical symmetry. As is well known, thesingular points of curvilinear coordinate systems, r = 0 in our case, can sometimesrequire special treatment to ensure that numerical solutions remain regular there.In critical collapse the highly dynamical nature of the solution near r = 0 mightnaturally be expected to exacerbate problems with regularity. In the work describedbelow we have paid special attention to the ability of our approach to both fullyresolve the near-critical configuration and maintain regularity of the solution at theorigin.The remainder of this paper is organized as follows: in Sec. 3.2 we review thegeneralized BSSN formulation and display the equations of motion for our modelsystem. Sec. 3.3 expands the discussion of the issue of regularity at the coordinatesingularity point, describes the numerical approach we have adopted, and providesdetails concerning the various tests and diagnostics we have used to validate ourimplementation. In Sec. 3.4 we present results computed using two distinct choicesfor the shift vector and provide conclusive evidence that the generalized BSSN for-mulation is capable of evolving in the critical regime in both cases. Sec. 3.5 containssome brief concluding remarks, and further details concerning the BSSN formal-ism in spherical symmetry and the scalar field equations of motion are includedin App. 3.6 and App. 3.7, respectively. We adopt units where the gravitationalconstant and the speed of light are both unity: G = c = 1.883.2. Equations of Motion3.2 Equations of MotionThe dynamical system we intend to study in the critical collapse regime is a real,massless scalar field, Ψ, self gravitating via Einstein’s equations,Gµν = 8πTµν . (3.3)Here, Tµν is the energy-momentum tensor associated with the minimally coupled Ψ:Tµν = ∇µ∇νΨ− 12gµν∇ηΨ∇ηΨ , (3.4)and the evolution of the scalar field is given by∇µ∇µΨ = 0 . (3.5)The time-development of the geometry is then given by recasting Einstein’s equa-tions as an evolution system based on the usual 3+1 expression for the spacetimemetric:ds2 = −α2dt2 + γij(dxi + βidt)(dxj + βjdt) . (3.6)Here, the 3-metric components, γij, are viewed as the fundamental dynamical geo-metrical variables and the lapse function, α, and shift vector, βi, which encode thecoordinate freedom of general relativity, must in general be prescribed independentlyof the equations of motion.3.2.1 Generalized BSSNWe now summarize the BSSN formulation of Einstein’s equations and describe howit can be adapted to curvilinear coordinates. Readers interested in additional detailsare directed to [28] for a more pedagogical discussion.In the standard ADM formulation [27, 87], the dynamical Einstein equations893.2. Equations of Motionare rewritten as evolution equations for the 3-metric and the extrinsic curvature{γij ,Kij}. The first difference between the BSSN formulation and the ADM de-composition is the conformal re-scaling of the ADM dynamical variables:γij = e4φγ˜ij , (3.7)Kij = e4φA˜ij +13γijK , (3.8)where eφ is the conformal factor, γ˜ij is the conformal metric, A˜ij is the conformallyrescaled trace-free part of the extrinsic curvature and K = γijKij is the trace ofthe extrinsic curvature. Here by fixing the trace of A˜ij, and the determinant of theconformal metric, the set of primary ADM dynamical variables transforms to thenew set:{γij ,Kij} → {φ, γ˜ij ,K, A˜ij} , (3.9)in the BSSN formulation.In the original BSSN approach, the conformal metric γ˜ij is taken to have deter-minant γ˜ = 1. However this choice is only suitable when we adopt coordinates inwhich the determinant of the flat-space metric reduces to unity. This is the case, ofcourse, for Cartesian coordinates but is not so for general curvilinear systems. Forinstance, the flat 3-metric in spherical coordinates:ds2 = dr2 + r2dθ2 + r2 sin2 θdφ2 , (3.10)has determinant γ˚ = r4 sin2 θ. Recently, Brown [66] has resolved this issue by intro-ducing a covariant version of the BSSN equations—the so-called generalized BSSNformulation, which we will hereafter refer to as G-BSSN—in which the primarydynamical variables are tensors so that the formulation can be adapted to non-Cartesian coordinate systems. In G-BSSN we no longer assume that the conformal903.2. Equations of Motion3-metric has determinant one. Rather, φ becomes a true scalar and for its dynamicsto be determined a prescription for the time evolution of the determinant of γ˜ij mustbe given. In the following this will be done by requiring that the determinant beconstant in time.Another main difference between the ADM decomposition and BSSN is that themixed spatial derivative terms occurring in the 3-Ricci tensor are eliminated throughthe definition of a new quantity, Γ˜k:Γ˜k ≡ γ˜ijΓ˜kij , (3.11)which becomes an additional, independent dynamical variable. Note that Γ˜i is nota vector as it is coordinate dependent. To extend this redefinition so that it is wellsuited for all coordinate choices, in G-BSSN we defineΛ˜k ≡ γ˜ij(Γ˜kij − Γ˚kij) = Γ˜k − Γ˚kij γ˜ij , (3.12)where Γ˚kij denotes the Christoffel symbols associated with the flat metric. Thisdefinition makes this so-called conformal connection, Λ˜i, a true vector and it becomesa primary dynamical variable in G-BSSN.We now summarize the G-BSSN equations, referring the reader to [88] for moredetails, including a full derivation. We begin by defining ∂⊥, the time derivativeoperator acting normally to the t = const. slices:∂⊥ ≡ ∂t −L~β , (3.13)where L~β denotes the Lie derivative along~β. We then have∂⊥φ = −16αK + σ16D˜kβk , (3.14)913.2. Equations of Motion∂⊥γ˜ij = −2αA˜ij − σ23A˜ijD˜kβk , (3.15)∂⊥K = −γijDjDiα+ α(A˜ijA˜ij + 13K2) + 4π(ρ+ S) , (3.16)∂⊥A˜ij = e−4φ [−DiDjα+ α(Rij − 8πSij)]TF+ α(KA˜ij − 2A˜ilA˜lj)− σ23A˜ijD˜kβk , (3.17)∂⊥Γ˜i = −2A˜ij∂jα+ γ˜lj∂j∂lβi+ 2α(Γ˜ijkA˜kj − 23γ˜ij∂jK + 6A˜ij∂jφ− 8πγ˜ijSj)+σ3[2Γ˜iD˜kβk + γ˜li∂l(D˜kβk)]. (3.18)Here, a superscript TF denotes the trace-free part (with respect to the 3-metricγij) of a tensor, and D˜i is the covariant derivative associated with the conformalmetric γ˜ij. Additionally, the quantity σ is an adjustable parameter that is discussedbelow and typically is either 0 or 1. Note that all the Lie derivatives in the G-BSSNequations operate on true tensors and vectors of weight 0. For instance,L~βA˜ij = βk∂kA˜ij + A˜ik∂jβk + A˜kj∂iβk . (3.19)Furthermore, in G-BSSN, rather than evolving (3.18), the redefined conformal con-nection, Λ˜i, is evolved via∂tΛ˜k = ∂tΓ˜k − Γ˚kij∂tγ˜ij , (3.20)where the time derivative ∂tγ˜ij is eliminated using (3.15). In equation (3.17), Rij923.2. Equations of Motiondenotes the 3-Ricci tensor associated with γij and can be written as the sumRij = Rφij + R˜ij , (3.21)where Rφij is given byRφij = −2D˜iD˜jφ− 2γ˜ijD˜kD˜kφ+ 4D˜iφD˜jφ− 4γ˜ijD˜kφD˜kφ , (3.22)and R˜ij is the 3-Ricci tensor associated with the conformal metric:R˜ij = − 12γ˜lm∂m∂lγ˜ij + γ˜k(i∂j)Γ˜k + Γ˜kΓ˜(ij)k+ γ˜lm(2Γ˜kl(iΓ˜j)km + Γ˜kimΓ˜klj). (3.23)The matter fields ρ, S, Si and Sij are defined byρ = nµnνTµν , (3.24)S = γijSij , (3.25)Si = −γijnµTµj , (3.26)Sij = γiµγiνTµν , (3.27)where nµ is the unit normal vector to the t = const. slices.As mentioned previously, we need to prescribe dynamics for the determinantof γ˜ij to have a complete set of equations of motion for the G-BSSN dynamicalvariables. One approach is to fix the determinant to its initial value by demanding933.2. Equations of Motionthat∂tγ˜ = 0 . (3.28)This is the so-called Lagrangian option and is associated with the choice σ = 1 inthe equations. Another option is to define the determinant to be constant along thenormal direction to the time slices, which can be implemented by requiring ∂⊥γ˜ = 0.This is usually referred to as the Lorentzian option, and is associated with the choiceσ = 0. Here we choose (3.28), i. e. σ = 1.Note that in the G-BSSN equations the divergence of the shift vector,D˜kβk =1√γ˜∂k(√γ˜βk) , (3.29)no longer reduces to ∂kβk since the determinant of the conformal metric γ˜ij isnot necessarily 1, but by virtue of the choice (3.28) is equal to that of the initialbackground flat metric in the chosen curvilinear coordinates.As usual, when setting initial data for any given evolution of the coupled Einstein-matter equations we must solve the Hamiltonian and momentum constraints. Interms of the G-BSSN variables these areH ≡ γ˜ijD˜iD˜jeφ − eφ8R˜+e5φ8A˜ijA˜ij− e5φ12K2 + 2πe5φρ = 0 , (3.30)Mi ≡ D˜j(e6φA˜ji)− 23e6φD˜iK − 8πe6φSi = 0 . (3.31)943.2. Equations of Motion3.2.2 G-BSSN in Spherical Symmetry and Gauge ChoicesIn spherical symmetry a generic form of the conformal metric γ˜ij is given byγ˜ij =γ˜rr(t, r) 0 00 r2γ˜θθ(t, r) 00 0 r2γ˜θθ(t, r) sin2 θ . (3.32)Similarly, a suitable ansatz for the traceless extrinsic curvature isA˜ij =A˜rr(t, r) 0 00 r2A˜θθ(t, r) 00 0 r2A˜θθ(t, r) sin2 θ . (3.33)The shift vector and Λ˜i have only radial components:βi = [β(t, r), 0, 0] , (3.34)Λ˜i =[Λ˜(t, r), 0, 0]. (3.35)Given (3.32-3.35), the G-BSSN equations become a set of first order evolution equa-tions for the 7 primary variables{φ(t, r), γ˜rr(t, r), γ˜θθ(t, r),K(t, r),A˜rr(t, r), A˜θθ(t, r), Λ˜(t, r)}.These are coupled to the evolution equation (3.5) for the scalar field and constrainedby the initial conditions (3.30–3.31). The explicit expressions for the full set ofequations of motion are given in App. 3.6.To fix the time slicing we implement a non-advective31 version of the 1+log31The terminology non-advective derives from the absence of an “advective” term, βj∂j , on the953.2. Equations of Motionslicing condition:32∂tα = −2αk . (3.36)for the spatial coordinates we either choose a zero shift:βi = 0 , (3.37)or use what we will term the gamma-driver condition:∂tβi = µΛ˜i − ηβi . (3.38)Here, µ and η are adjustable parameters which we set to µ = 3/4 and η ≃1/(2MADM), where MADM is the total mass of the system measured at infinity(see Sec. 3.3.4). We emphasize that (3.38) is not the usual Gamma-driver equationused in the standard BSSN approach:∂tβi = µΓ˜i − ηβi , (3.39)but since it is a natural extension of the above to the G-BSSN case we have optedto use the same nomenclature. In the rest of this paper, we frequently refer tothe shift vector evolved via (3.38) as βG. Explicitly, in spherical symmetry βG isdefined by∂tβG(t, r) = µΛ˜(t, r)− ηβG(t, r) . (3.40)left hand side of equations (3.36,3.38). we note that we also experimented with the advectiveversions of the equations. the results were very similar to those for the non-advective case; inparticular, near-critical solutions exhibiting echoing and scaling could also be obtained.32The reader can easily check that in the case of zero shift, the lapse choice given by (3.36)combined with (3.14) implies ∂t(α − 12φ) = 0. In Cartesian coordinates 12φ = ln γ, so this lastequation gives α − ln γ = c(~x), where the function c(~x) is time independent. the choice c(~x) = 1then yields an algebraic expression for the lapse, α = 1+ln γ, which is the origin of the terminology“1+log slicing”.963.3. Numerics3.3 NumericsWe use a second order finite differencing method to discretize equations (3.14-3.17)and (3.20). Further, the equations of motion are transformed to a compactifiedradial coordinate that we denote by r˜ and which is defined in terms of the originalcoordinate r byr = er˜ − eδ + R∞R∞ − r˜ −R∞R∞ − δ , (3.41)where δ and R∞ are parameters with typical values δ ≃ −12 and R∞ ≃ 3. It isstraightforward to verify the following:1) the radial domain r = (0,∞) maps to the computational domain r˜ = (δ,R∞),2) the derivative dr/dr˜ decreases toward the origin (r˜ ≃ δ), so that a uniform gridon r˜ is a non-uniform grid on r with approximately 103 times more resolution closeto the origin relative to the outer portion of the solution domain, r˜ ≃ 2 (r ≃ 10),where the support of the scalar field is initially concentrated,3) the parameter δ can be used to adjust the resolution near the origin; specifically,decreasing δ increases the resolution near r = 0. For notational simplicity, however,in the following we omit the explicit dependence of the fields on r˜ and denote thespacetime dependence of any dynamical variable X as previously: X (t, r(r˜)) ≡X(t, r).We use a finite difference grid that is uniform in r˜ and analytically transformall r-derivative terms in the equations of motion to their r˜-coordinate counterpartsprior to finite-differencing.We also developed a Maple-based toolkit [89] that automates the process ofdiscretizing an arbitrary derivative expression. This toolkit handles boundary con-ditions and generates a point-wise Newton-Gauss-Seidel solver in the form of Fortranroutines for a given set of time dependent or elliptic PDEs . The calculations in thispaper were all carried out using this infrastructure.973.3. Numerics3.3.1 InitializationThe matter content is set by initializing the scalar field to a localized Gaussian shell:Ψ(0, r) = p exp(−(r − r0)2σ2r), (3.42)where p, r0 and σr are parameters. Note that here r is the non-compactified radialcoordinate which is related to the compactified coordinate r˜ via (3.41). A typicalinitial profile for the scalar field in our calculations has σr ≃ 1, r0 ≃ 10, and p oforder 10−1. We use the overall amplitude factor p as the tuning parameter to findcritical solutions. We initialize the conformal metric (3.32) to the flat metric inspherical symmetry,γ˜rr(0, r) = γ˜θθ(0, r) = 1 , (3.43)and initialize the lapse function to unity,α(0, r) = 1 . (3.44)We also demand that the initial data be time-symmetric,A˜rr(0, r) = A˜θθ(0, r) = K(0, r) = 0 , (3.45)β(0, r) = Λ˜(0, r) = 0 , (3.46)∂tΨ(t, r)|t=0 = 0 , (3.47)which means that the momentum constraint (3.31) is trivially satisfied. This leavesthe Hamiltonian constraint (3.30) which is solved as a two-point boundary value983.3. Numericsproblem for the conformal factor at the initial time,ψ(r) ≡ eφ(0,r) . (3.48)The outer boundary condition for ψ,ψ(r)|r=∞ = 1 , (3.49)follows from asymptotic flatness, while at r = 0 we have∂rψ(r)|r=0 = 0 (3.50)since ψ(r) must be an even function in r for regularity at the origin.3.3.2 Boundary ConditionsDue to the fact that the metric has to be conformally flat at the origin we haveγ˜rr(t, 0) = γ˜θθ(t, 0) . (3.51)Further, since we are using the Lagrangian choice, σ = 1, the determinant of γ˜ijmust at all times be equal to its value at the initial time, soγ˜rrγ˜2θθ = 1 . (3.52)From these two results we haveγ˜rr(t, 0) = γ˜θθ(t, 0) = 1 . (3.53)Using (3.53) and (3.15) it is then easy to see that we must also haveA˜rr(t, 0) = A˜θθ(t, 0) = 0 . (3.54)993.3. NumericsAs is usual when working in spherical coordinates, many of the boundary con-ditions that must be applied at r = 0 follow from the demand that the solution beregular there. Essentially, the various dynamical variables must have either even orodd “parity” with respect to expansion in r as r → 0. Variables with even parity,typically scalars or diagonal components of rank-2 tensors, must have vanishing ra-dial derivative at r = 0, while odd parity functions, typically radial components ofvectors, will themselves vanish at the origin.Applying these considerations to our set of unknowns we find∂rγ˜rr(t, r)|r=0 = ∂rγ˜θθ(t, r)|r=0 = 0 , (3.55)∂rA˜rr(t, r)|r=0 = ∂rA˜θθ(t, r)|r=0 = 0 , (3.56)β(t, 0) = Λ˜(t, 0) = 0 , (3.57)∂rK(t, r)|r=0 = ∂rφ(t, r)|r=0 = ∂rΨ(t, r)|r=0 = 0 . (3.58)We use equations (3.53,3.54,3.57) to fix the values of the functions at the origin anda forward finite-differencing of (3.58) to update K, φ and Ψ at r = 0. Further, weapply a forward finite-differencing of (3.55,3.56) to update the values of the functionat the grid point next to the origin. The 1+log condition (3.36) can be used directlyat r = 0. Again, we emphasize that all of the r-derivative terms of the boundaryconditions described above are analytically transformed to the numerical coordinate,r˜, before the equations are finite-differenced.Since we are using compactified coordinates, all the variables are set to their flat1003.3. Numericsspacetime values at the outer boundary r =∞:γ˜rr = γ˜θθ = eφ = α = 1 at : (t,∞) , (3.59)A˜rr = A˜θθ = K = Λ˜ = β = Ψ = 0 at : (t,∞) . (3.60)Here, we emphasize that spatial infinity, r =∞, corresponds to the finite compact-ified (computational) coordinate point r˜ = R∞.3.3.3 Evolution Scheme and RegularityWe implemented a fully implicit, Crank-Nicolson [90] finite differencing scheme toevolve the system of G-BSSN equations. The precise form of the continuum equa-tions used is given in App. 3.6 and all derivatives, both temporal and spatial, wereapproximated using second-order-accurate finite difference expressions.During an evolution the correct limiting behaviour of the spatial metric compo-nents must be maintained near r = 0 to ensure a regular solution. For example, thelimiting values of the conformal metric components γ˜rr and γ˜θθ are given byγ˜rr(t, r) = 1 +O(r2) , (3.61)γ˜θθ(t, r) = 1 +O(r2) . (3.62)If the discrete approximations of the metric functions do not satisfy these conditions,then irregularity will manifest itself in the divergence of various expressions such asthe Ricci tensor component (3.95)Rrr = 2γ˜rr − γ˜θθr2γ˜θθ+ · · · , (3.63)which should converge to a finite value at the origin if conditions (3.61,3.62) hold.1013.3. NumericsOne approach to resolve potential regularity issues is to regularize the equations[65, 88, 91], by redefining the primary evolution variables, so that the equationsbecome manifestly regular at the origin. Another approach is to use implicit orpartially implicit methods [92]. As recently shown by Montero and Cordeo-Carrion[93], such schemes can yield stable evolution without need for explicit regularization.Baumgarte et al. [94] also adopted a similar approach—using a partially implicitscheme without regularization—in an implementation of the G-BSSN formulationin spherical polar coordinates.As mentioned, our implementation is fully implicit and we have also found thatour generalized BSSN equations can be evolved without any need for regulariza-tion at the origin, even in strong gravity scenarios where the spacetime metric hassignificant deviations from flatness near the origin.That said, we also experimented with other techniques aimed at improving reg-ularity. For example, using the constraint equation (3.52) and the fact that A˜ij istrace-free,A˜rrγ˜rr+ 2A˜θθγ˜θθ= 0 , (3.64)we can compute γ˜θθ and A˜θθ in terms of γ˜rr and A˜rr, respectively, rather thanevolving them. However, when we did this we found no significant improvement inregularity relative to the original scheme.Finally, to ensure our solutions remain smooth on the scale of the mesh we usefourth order Kreiss-Oliger dissipation [95] in the numerical solution updates.3.3.4 TestsThis section documents various tests we have made to validate the correctness ofour numerical solver as well as the consistency of the finite-differencing methodused to evolve the system of G-BSSN equations. We use a variety of diagnostictools, including monitoring of the constraint equations, convergence tests of theprimary dynamical variables, and a direct computation to check if the metric and1023.3. NumericsFigure 3.1: Results from various tests that verify the accuracy and consistency of ournumerical solver and the finite differencing method used to integrate the equations.(a) The evolution of the l2-norm (RMS value) of the Hamiltonian constraint. Thenorm is plotted for 3 different resolutions h, h/2 and h/4 corresponding to Nr = 512,1024 and 2048, respectively. The data for the Nr = 1024 and Nr = 2048 curveshave been rescaled by factors of 4 and 16, respectively, and the overlap of the threelines thus signals the expected second order convergence to zero of the constraintdeviation. We observe similar convergence properties for the momentum constraintas well as the constraint equation (3.12) for Λ˜i, and the constraint (3.64) for thetrace of A˜ij . Additionally, since the operator used to evaluate the residual of theHamiltonian constraint is distinct from that used in the determination of the initialdata, the test also validates the initial data solver. (b) Conservation of the ADMmass during the evolution of strong initial data. Here the deviation of the mass fromits time average is plotted for 3 different resolutions. Higher resolution values haveagain been rescaled so that overlap of the curves demonstrates O(h2) convergenceto 0 of the deviation of the total mass.1033.3. NumericsFigure 3.1: (c) The convergence factor defined in (3.68) for three of the primaryBSSN variables: γ˜rr, K, and A˜θθ. In the limit h → 0 we expect all curves to tendto the constant 4. The plot thus provides evidence for second order convergence ofall of the values throughout the evolution. All of the other primary BSSN variablesas well as the dynamical scalar field quantities demonstrate the same convergence.(d) Direct verification that the metric found by numerically solving the BSSN equa-tions satisfies Einstein’s equations in their covariant form. Here the tr componentof the residual Eµν defined in (3.72) is plotted for 3 different resolutions. Once more,higher resolution values have been rescaled so that overlap of the curves signals theexpected O(h2) convergence of the residuals to 0. All of the plots correspond toevolution of strong subcritical initial data with 1+log slicing. For (a) and (b) theshift vector was set to 0, while in (c) and (d) it was evolved using the Gamma-drivercondition (i.e. β = βG).matter fields calculated via the G-BSSN formulation satisfy the covariant form ofEinstein’s equations. All of the calculations were performed using the 1+log slicingcondition (3.36) and either β = 0 or β = βG where βG satisfies the Gamma-drivercondition (3.40).Constraints and Conserved QuantitiesWe monitor the evolution of the constraint equations (3.30,3.31) during a strongly-gravitating evolution where the nonlinearities of the equations are most pronounced.As demonstrated in Fig. 3.1 (a), at resolutions typical of those used in our study,the Hamiltonian constraint is well preserved during such an evolution and, impor-tantly, the deviations from conservation converge to zero at second order in the meshspacing as expected.The total mass-content of the spacetime seen at spatial infinity (the ADM mass)is a conserved quantity. Here, using the G-BSSN variables the Misner-Sharp massfunction is given byM(r) =rγ˜1/2θθ e2φ2[1− γ˜θθγ˜rr(1 + r∂r γ˜θθ2γ˜θθ+ 2r∂reφeφ)2]. (3.65)1043.3. NumericsThe total mass, MADM, can be evaluated at the outer boundary,MADM ≡M(r =∞) . (3.66)The deviation of the total mass from its time average is plotted in Fig. 3.1(b); asthe resolution of the numerical grid increases the variations converge to zero in asecond order fashion.Convergence TestAs mentioned in Sec. 3.3.1 and Sec. 3.3.3, we implemented our code using secondorder finite differencing of all spatial and temporal derivatives. Denoting any con-tinuum solution component by q(t,X), where X is the spatial coordinate, and adiscrete approximation to it computed at finite difference resolution, h, by qh(t,X),to leading order in h we expectqh(t,X) = q(t,X) + h2e2[q](t,X) + . . . . (3.67)Fixing initial data, we perform a sequence of calculations with resolutions h, h/2and h/4 and then compute a convergence factor, C(t; q), defined byC(t; q) =||qh(t,X) − qh/2(t,X)||2||qh/2(t,X) − qh/4(t,X)||2, (3.68)where || · ||2 is the l2 norm, i.e. the root mean square (RMS) value. It is straightfor-ward to argue from (3.67) that, for sufficiently small h, C(t, q) should approach 4 ifthe solution is converging at second order. The values of the convergence factor for aselection of dynamical variables are plotted for a strong-data evolution in Fig. 3.1(c)and provide clear evidence that the solution is second-order convergent throughoutthe time evolution.1053.3. NumericsDirect Validation via Einstein’s EquationsA direct method to test the fidelity of our numerical solver involves the evaluationof a residual based on the covariant form of Einstein’s equations. We start with areconstruction of the four-dimensional metric in spherical symmetry,ds2 = (−α2 + β2a2)dt2 + 2a2βdtdr + a2dr2 + r2b2dΩ2 , (3.69)using the primary G-BSSN variables, γ˜ij and φ. In particular, a and b are simplygiven bya(t, r) = e4φ(t,r)γ˜rr(t, r) , (3.70)b(t, r) = e4φ(t,r)γ˜θθ(t, r) . (3.71)We then check to see if the metric (3.69) satisfies the covariant Einstein equa-tions (3.3) to the expected level of truncation error. Specifically, defining the residualEµν ≡ Gµν − 8πT µν , (3.72)and replacing all derivatives in Gµν with second order finite differences, we expectEµν to converge to zero as O(h2) as h → 0.33 Precisely this behaviour is shownin Fig. 3.1(d). This is a particularly robust test of our implementation since thenon-trivial components of the covariant Einstein equations are quite complicatedand, superficially at least, algebraically independent of the BSSN equations. For33Although it is not crucial for the usefulness of this test, we discretize the Eµν using a differencescheme that is distinct from the one used in the main code.1063.3. Numericsinstance, the tr component of the residual (3.72) is given byEtr =2βrα2(∂raa− 2∂rbb+∂rαα)+2βα2(−∂2r bb+∂ra∂rbab− (∂rΨ)22+∂rα∂rbαb)+2α2(∂t∂rbb+∂rΨ∂tΨ2− ∂ta∂rbab− ∂tb∂rααb)+2rα2(−∂taa+∂tbb)(3.73)and depends on all of the dynamical variables of the system. The observed conver-gence of the residual is only plausible if 1) our G-BSSN equations (3.14-3.18) havebeen correctly derived from the covariant Einstein equations, 2) we have discretizedthe geometric and matter equations properly, and 3) we have solved the full set ofdiscretized equations correctly.3.3.5 Finding Black Hole Threshold SolutionsThe strength of the initial data can be set by adjusting the amplitude of the scalarfield, p, in (3.42). For weak enough initial data (small enough p), the matter shellwill reach the origin and then disperse, with the final state being a flat spacetimegeometry. Sufficiently strong initial data, on the other hand (large enough p), resultsin a matter concentration in the vicinity of the origin which is sufficiently self-gravitating that a black hole forms. Using a binary search, we can find the thresholdinitial data, defined by p = p⋆, for which p < p⋆ results in dispersal while p > p⋆yields black hole formation. At any stage of the calculation, the binary search isdefined by two “bracketing” values, pl and ph, such that evolutions with p = pl andp = ph result in dispersal and black hole formation, respectively. It is convenientto define the amount of parameter tuning that has occurred by the dimensionlessquantityδp ≡ ph − plpl. (3.74)1073.4. ResultsThe dispersal case can be detected easily as the scalar field leaves the vicinityof the origin and the geometry approaches flat spacetime. To detect black holeformation, we use an apparent horizon finder to locate a surface r = const. on whichthe divergence of the outgoing null rays vanishes. We first define the divergencefunctionΘ = qµν∇µkν , (3.75)where qµν is the induced metric on the constant r surface. In spherical symmetrywith metric (3.69) we haveqµν = diag(0, 0, r2b2, r2b2 sin2 θ), (3.76)where kµ is the null outgoing vector given bykµ =1√2[aβ − α, a, 0, 0] . (3.77)Therefore, (3.75) becomesΘ =√2rb(rα∂t(b) +(1a− βα)∂r(rb)). (3.78)The formation of an apparent horizon34 is signaled by the value of the function Θcrossing zero at some radius implying that the spacetime contains a black hole. Wenote that since the focus of our work was on the critical (threshold) solution wemade no effort to continue evolutions beyond the detection of trapped surfaces.3.4 ResultsIn this section we describe results from two sets of numerical experiments to studythe efficacy of the G-BSSN formulation in the context of critical collapse. Again,34Technically a marginally trapped surface—the apparent horizon being the outermost of these.1083.4. Resultsour calculations use the standard 1+log slicing condition for the lapse, and a shiftwhich is either zero or determined from the Gamma-driver condition.3.4.1 Zero ShiftWe first perform a collection of numerical experiments where the shift vector is setto zero. As described in Sec. 3.3.5, in principle we can find the black hole thresholdsolution p ≃ p⋆ using a binary search algorithm which at any stage is defined bytwo values pl and ph, with pl < p < ph, and where pl corresponds to dispersal (weakdata) while ph corresponds to black hole formation (strong data).As discussed in the introduction, the massless scalar collapse model has a verywell-known critical solution, and we summarize the features most relevant to ourstudy here. The threshold configuration is discretely self-similar with an echoingexponent measured from the first calculations to be ∆ ≈ 3.44 [4]. Following theoriginal studies, Gundlach [67] showed that the construction of the precisely dis-cretely self-similar spacetime could be posed as an eigenvalue problem, the solutionof which led to the more accurate value ∆ = 3.4439 ± 0.0004. This estimate wassubsequently improved by Martin-Garcia and Gundlach to ∆ = 3.445452402(3) [72].The original calculations determined a value γ ≈ 0.37 for the mass-scaling ex-ponent [4]; further work based on perturbation theory gave γ ≈ 0.374 [68, 71]. Hereit is important to note that, as pointed out independently by Gundlach [68] andHod and Piran [69], the simple power law scaling (3.1) gets modified for discretelyself-similar critical solutions tolnM = γ ln |p− p⋆|+ c+ f (γ ln |p− p⋆|+ c) , (3.79)where f is a universal function with period ∆ and c is a constant depending on theinitial data. This results in the superposition of a periodic “wiggle” in the otherwiselinear scaling of lnM as a function of ln |p− p⋆|.Finally, Garfinkle and Duncan [70] pointed out that near-critical scaling is seen1093.4. ResultsFigure 3.2: Echoing behaviour in the scalar field for a marginally subcritical evo-lution with δp ≈ 10−12. The main plot displays the central value of the scalar fieldversus a logarithmically scaled time parameter, ln(Tf−T ), where T is central propertime and Tf is the approximate value of that time when near-critical evolution ceasesand the total dispersal of the pulse to infinity begins. This particular scaling is cho-sen solely to more clearly demonstrate the evolution of the central value of Ψ duringthe critical phase through to dispersal. Note that our choice of abscissa means thatevolution proceeds from right to left. The inset also plots Ψ at r = 0 but now inthe “natural” logarithmic time coordinate τ ≡ ln(T ⋆−T ) where T ⋆ is the “accumu-lation time” at which the solution becomes singular and which has been estimatedbased on the positions of the extrema in Ψ. The amplitude of the scalar field atthe origin oscillates between (−0.61, 0.61), consistent with the calculations reportedin [4]. The data yield an echoing exponent of ∆ = 3.43± 0.02 which is in agreementwith the value ∆ = 3.445452402(3) Martin-Garcia and Gundlach have computed bytreating the computation of the precisely-critical solution as an eigenvalue problem[72].1103.4. ResultsFigure 3.3: The maximum central value, Rmax, of the four-dimensional Ricci scalar,R, attained during subcritical evolution as a function of the logarithmic distanceln |p−p⋆| of the tuning parameter from the critical value. As first observed in [70] theRicci scalar scales as R ∼ |p−p⋆|−2γ , where γ is the universal mass-scaling exponentin (3.1). The value γ = 0.38± 0.01 computed via a least squares fit is in agreementwith the original calculations [4] as well as many other subsequent computations.We note that the oscillations of the data about the linear fit are almost certainlygenuine, at least in part. As discussed in the text, we expect a periodic wiggle inthe data with period ∆/(2γ) ≈ 4.61. Performing a Fourier analysis of the residualsto the linear fit we find a peak at about 4 with a bandwidth of approximately 1,consistent with that expectation. As described in more detail in the text, althoughwe have data from computations with ln |p− p⋆| < −25, we do not include it in thefit. The naive method we use to estimate p⋆ means that the relative uncertainty inp − p⋆ grows substantially as p → p⋆ so that inclusion of the data from the mostnearly-critical calculations will corrupt the overall fit.1113.4. Resultsin physical quantities other than the mass and, dependent on the quantity, in thesubcritical as well as supercritical regime. In particular they argued that in sub-critical evolutions the maximum central value, Rmax, of the four-dimensional Ricciscalar, R, defined byRmax ≡ maxtR(0, t) , (3.80)should satisfy the scalingRmax ∼ |p− p⋆|−2γ , (3.81)where the factor −2 in the scaling exponent can be deduced from the fact that thecurvature has units of length−2. For the discretely self-similar case this scaling lawis also modulated by a wiggle with period ∆/(2γ), which for the massless scalarfield is about 4.61.Using initial data given by (3.42) we tune p so that it is close to the criticalvalue: typically this involves reducing the value of δp defined by (3.74) so that it isabout 10−12, which is a few orders of magnitude larger than machine precision. Ourimplementation includes code that actively monitors the dynamical variables forany indications of coordinate singularities or other pathologies which could causethe numerical solver to fail. Provided that such pathologies do not develop, weexpect to observe features characteristic of critical collapse—discrete self-similarityand mass scaling in particular—to emerge as p→ p⋆.One way the discrete self-similarity of the critical solution is manifested is as asequence of “echoes”—oscillations of the scalar field near the origin such that aftereach oscillation the profile of the scalar field is repeated but on a scale exp(∆) smallerthan that of the preceding echo (see Eq. (3.2)). The oscillations are similarly periodicin the logarithmic time scale ln(T ⋆ − T ), where T is the proper time measured atthe origin,T (t) ≡∫ t0α(tˆ, 0) dtˆ , (3.82)and T ⋆ is the accumulation time at which the singularity forms (always at r = 0).1123.4. ResultsFigure 3.4: Discrete self-similarity of the geometry of spacetime in the black holethreshold evolution previously discussed in Fig. 3.2. Here, the G-BSSN variable φ isplotted as a function of the computational radial coordinate r˜ at the accumulationtime t⋆. Note that from (3.83) φ measures the deviation of the determinant of the 3-metric from that of a flat metric. The inset graph is the radial derivative of φ scaledby√r to highlight the formation of fine structure in the geometry of the criticalsolution. The approximate periodicity of√rφ′ in ln(r) (modulo an overall varyingscale) provides weak evidence that the coordinate system used in the calculationadapts to the self-similarity of the critical solution.1133.4. ResultsFigure 3.5: Snapshots of radial mass density for a marginally subcritical calculation(δp ≈ 10−12, Nr = 2048). Plotted is dm/dr = r2ρ(t, r) where ρ is defined by (3.24).In this calculation β = 0 so we also have dm/dr = T tt. As the solution evolves,development of echos is clearly seen. In the final frame, which is at an instantt = 15.8 that is close to the accumulation time t⋆, we observe 4 echos. Note thatwe do not count the tall thin peak at the extreme left nor the first two peaks onthe right as echos. The skinny peak will develop into an echo as p is tuned closerto p⋆. The two peaks on the right account for the bulk of the matter and representthe part of the initial pulse that implodes through the origin and then disperses“promptly”, i.e. without participating in the strongly self-gravitating dynamics. Acorresponding plot for an evolution far from criticality would contain only thosetwo peaks. Note that the first three plots use the computational coordinate r˜ toprovide a sense of the actual numerical calculation, while the last plot uses ln(r)in order to best highlight the discrete self-similarity of the threshold solution. Asis the case for the data plotted in the inset of the previous figure, the approximateperiodicity of the mass density in ln(r) suggests that the coordinates are adaptingto the self-symmetry of the critical spacetime.1143.4. ResultsFurthermore, viewed at the origin, the oscillations of the scalar field occur at afixed amplitude of about 0.61 (with our units and conventions for the Einstein’sequations). As shown in Fig. 3.2, when we tune the initial data to the critical value,the central value of scalar field exhibits oscillatory behaviour and the amplitude isclose to the expected value. The anticipated periodicity in logarithmic time is alsoapparent with a measured ∆ = 3.43 ± 0.02, in agreement with previous results.We thus have strong evidence that the evolution has indeed approached the criticalregime and that the measured oscillations are true echos rather than numericalartifacts.Evidence that our code correctly captures the expected critical scaling behaviour (3.81)of Rmax is presented in Fig. 3.3. We find γ = 0.38 ± 0.01, consistent with previouscalculations. We note that we can measure scaling from our computations up toln |p − p⋆| = −29 (or |p − p⋆| ≈ 10−13). However, in Fig. 3.3 we have excluded thelast few values closest to the critical point from both the plot and the linear fit:specifically, we truncate the fit at ln |p − p⋆| = −25. The rationale for this is thatwe use the largest subcritical value of p as an approximation to the critical valuep⋆ rather than, for example, implementing a multi-parameter fit that includes p⋆ asone of the parameters. Our estimate of p⋆ thus has an intrinsic error of e−29 ≈ 10−13and by fitting to data with ln |p− p⋆| ≥ −25 we render the error in the p⋆ estimateessentially irrelevant. We note that consistent with the early observations of the ro-bustness of mass scaling in the model [4], measuring the exponent γ can be achievedby moderate tuning, in this case ln |p − p⋆| ≈ −9, (i.e. δp ≈ 10−3). However, tobe able to observe the echoing exponent (the oscillations around the fitting line, forexample) we need to tune much closer to the critical value.The echoing behaviour of the critical solution is also reflected in the geometryof spacetime and the matter variables other than the scalar field. Fig. 3.4 shows theradial profile of the G-BSSN variable φ at an instant close to the accumulation timeT ⋆. As seen in this plot, fine structure develops in the function in the near-critical1153.4. Resultsregime. Observe that from the definition (3.7) and the choice (3.28), the scalar φ isthe ratio of the determinant of the 3-metric, γ, to the determinant of the flat metric,γ˚:φ =112ln(γ/˚γ) . (3.83)The radial matter density, dm/dr = r2ρ, is a convenient diagnostic quantity forviewing near-critical evolution. Snapshots of this function from a typical marginallysubcritical calculation are shown in Fig. 3.5: the echoing behavior is clearly evidentin the sequence. The number of echos is dependent on the degree to which thesolution has been tuned to criticality. In this case, where δp = 10−12, we expect andsee about 4 echos (last frame of the figure). Here we note that each of the echos indm/dr corresponds to half of one of the scalar field oscillations shown in Fig. 3.2(where the inset shows about 212 full cycles).Fig. 3.6(a) plots the central matter density ρ(t, 0) for a marginally supercriticalcalculation. In accord with the self-similar nature of the near-critical solution, thecentral density grows exponentially with time. Fig. 3.6(b) is a snapshot of theextrinsic curvature at the critical time t ≈ t⋆ while Fig. 3.6(c) shows the dynamicsof the central value of the lapse function and compares it with α from the calculationsperformed with β = βG described in the next section. Fig. 3.6(d) displays the profileof the lapse at the critical time.We note that we have not fully resolved the critical behaviour in the sense ofhaving tuned p to the limit of machine precision, δp ≈ 10−16, which would captureroughly 2 additional echos in the threshold solution (one full echo in the scalar field).In principle, by setting Nr sufficiently large we could almost certainly do so sincethere are no indications that our method would break down at higher resolutionand closer to criticality. However, we estimate that the required compute timefor a complete critical search would increase from weeks to several months and wehave thus not done this. Ultimately, a more effective approach to enhancing theresolution would be to incorporate a technique such as adaptive mesh refinement1163.4. ResultsFigure 3.6: Profiles of matter and geometry variables from strongly gravitating,near-critical evolutions where the echoing behaviour emerges. Results were com-puted using 1+log slicing and zero shift, except for the dashed curve in (c) whereβ = βG. (a) Central energy density, ρ(T, 0), as a function of proper central time,T , and in logarithmic scale for a supercritical evolution. The density oscillates andgrows exponentially as the system approaches the critical solution and then eventu-ally collapses to form a black hole. (b) Profile of the extrinsic curvature, K(t⋆, r)—scaled by r1/2 in order to make the echoing behaviour more visible—where t⋆ denotesa time very close to the accumulation time. The evolution is marginally subcriticalin this case.1173.4. ResultsFigure 3.6: (c) Central value of the lapse function, α, during subcritical evolutionswith β = 0 (solid) and β = βG (dashed). The plots use a logarithmically transformedproper time variable, − ln(Tf − T ), where Tf is the approximate time at which thefinal dispersal of the pulse from the origin begins. In both cases α exhibits echoingand there is no evidence of pathological behaviour, such as the lapse collapsing orbecoming negative. The close agreement of α for the two choices of β indicates thatthe time slicing varies little between the two coordinate systems. Note that there arethree extra oscillations for the β = βG case, in the time interval − ln(Tf −T ) & 4.5.These are spurious and due to a lack of finite-difference resolution; there are only6 time steps in each oscillation. (d) Radial profile α(t⋆, r) at a time t = t⋆ whichis close to the accumulation time and when the self-similarity and echoing in thespacetime geometry is apparent.into our solver.The results displayed in Figs. 3.2–3.6 provide strong evidence that the coordinatesystem consisting of 1+log lapse and zero shift remains non-singular in the criticalregime, at least for the range of scales probed for δp ≈ 10−12. Additionally, theapproximate periodicity in ln(r) that can be seen, for example, in√rφ′ (Fig. 3.4)and dm/dr (Fig. 3.5) suggests that the coordinates may be adapting to the self-similarity. Whether or not this is actually the case is a matter requiring furtherstudy.3.4.2 Gamma-driver ShiftWe now briefly report on experiments similar to those of the previous section butwhere the shift was evolved with the Gamma-driver condition (3.38). A principalobservation is that this gauge also facilitates near-critical evolutions with resultssimilar to the β = 0 choice. In particular, we are again able to observe all of thecharacteristics of the black hole threshold solution.The gauge condition (3.38) acts as a damping factor for the conformal connec-tion, Λ˜i, and we would therefore expect to observe a significant change in the profileof Λ˜i at threshold relative to the zero-shift case. This expectation is borne out bythe comparison illustrated in Figs. 4.1 (a) and (b). When β = 0, Λ˜i diverges as1183.4. ResultsFigure 3.7: Profiles of various G-BSSN variables from marginally subcritical evo-lutions. (a) Profile of the conformal connection Λ˜ as computed with β = 0 and ata time t⋆ close to the accumulation time. Note that the function has been scaledby r and in fact diverges like 1/r. (b) Profile of Λ˜ as computed with β = βG, againat a moment close to the accumulation time. Here the 1/r growth seen when thecondition β = 0 is adopted is absent. (c) Profile of the central spatial derivativeof the shift vector, β′(t, 0), as computed with β = βG. As the echos develop closerto the origin, β′ increases and presumably will diverge in the continuum, precisely-critical limit. (d) Time development of the l2-norm of the extrinsic curvature duringsubcritical evolutions for both the β = 0 and β = βG calculations. In both casesthe extrinsic curvature develops a divergent profile near r = 0 in the critical regime.1193.5. Conclusion1/r close to the origin while it appears to have finite amplitude for β = βG. Wefind that the shift develops very sharp oscillations near the origin; some typical be-haviour can be seen in the plot of β′(t, 0) shown in Fig. 4.1(c). We believe thatthese oscillations are genuine and our expectation is that β′(t, 0) will diverge in theprecise critical limit. Further, we observe that the oscillations can create numericalartifacts and generally require higher resolution relative to the β = 0 case, as wellas dissipation, to be controlled. Indeed, when using the Gamma-driver condition wefind that Kreiss/Oliger dissipation is crucial to suppress unresolved high frequencyoscillations close to the origin. Fig. 4.1 (d) shows the growth in the norm of theextrinsic curvature during a subcritical evolution. The norm of K does not exhibitany significant difference for the two choices of the shift.As was the case for the β = 0 calculations, the results shown in Figs. 3.6 and3.7 strongly suggest that the combination of 1+log slicing and Gamma driver shiftprovides a coordinate system which is adequate for computing the near-critical so-lution. In addition, the approximate periodicity seen in Figs. 3.6(b), 3.6(c), 3.7(a)and 3.7(b) suggest that this gauge may also be adapting to the self-symmetry.3.5 ConclusionWe have described a numerical code that implements a generalized BSSN formula-tion adapted to spherical symmetry. Using standard dynamical coordinate choices,including 1+log slicing and a shift which either vanished or satisfied a Gamma-driver condition, we focused specifically on the applicability of the formulation andthe gauge choices to studies of type II critical phenomena. As a test of the approachwe revisited the model of massless scalar collapse, where the properties of the criticalsolution are very well known from previous work. For both choices of the shift, wefound that our code was able to generate evolutions that were very close to criti-cality so that, in particular, we could observe the expected discrete self-similarityof the critical solution. To our knowledge, this is the first fully evolutionary imple-1203.5. Conclusionmentation of a hyperbolic formulation of Einstein’s equations that has been able tounequivocally resolve discrete self-similarity in type II collapse. Furthermore, mea-sured properties from near-critical solutions, including the mass-scaling and echoingexponents, are in agreement with previous work. Our results strongly suggest thatthe G-BSSN formulation, in conjunction with standard dynamical coordinate condi-tions, is capable of evolving the spacetime near criticality without the developmentof coordinate pathologies. There is also some evidence that both gauges adapt tothe self-similarity, but we have not yet studied this issue in any detail.We found that certain of the primary G-BSSN variables diverge as the criticalsolution is approached: this is only to be expected since the precisely critical solutioncontains a naked singularity. Dealing with such solution features in a stable andaccurate manner presents a challenge for any code and in our case we found that acombination of a non-uniform grid and Kreiss/Oliger dissipation was crucial. Ouruse of a time-implicit evolution scheme may have also been important although wedid not experiment with that aspect of our implementation. However, we suspectthat the implicit time-stepping helped maintain regularity of the solutions nearr = 0, as other researchers have found.Given the success of the G-BSSN approach, it is natural to consider its gen-eralization and application to settings with less symmetry, but where curvilinearcoordinates are still adopted. In particular, one axisymmetric problem that has yetto be resolved is the collapse of pure gravity waves. This scenario arguably pro-vides the most fundamental critical phenomena in gravity as the behaviour mustbe intrinsic to the Einstein equations, rather than being dependent on some mattersource. Critical collapse of gravitational waves–with mass scaling and echoing—wasobserved by Abrahams and Evans over 20 years ago [76]. However, their originalresults have proven very difficult to reproduce (or refute) [77–80, 83]. We referthe reader to the recent paper by Hilditch et al. [83] for detailed discussions con-cerning some apparent inconsistencies among the follow-up studies, as well as the1213.6. BSSN in Spherical Symmetrychallenges and complications involved in evolving various types of nonlinear gravi-tational waves. We are currently extending the methodology described above to theaxisymmetric case with plans to use the resulting code to study vacuum collapse.Results from this undertaking will be reported in a future paper.3.6 BSSN in Spherical SymmetryIn this appendix, we provide the explicit expressions of the G-BSSN evolution equa-tions in spherical symmetry.The evolution equations (3.14-3.15) for φ and the components of the conformalmetric γ˜ij simplify to∂tφ =16αK + β∂rφ+ σ16B , (3.84)∂tγ˜rr = −2αA˜rr + β∂rγ˜rr + 2γ˜rr∂rβ − σ23γ˜rrB , (3.85)∂tγ˜θθ = −2αA˜θθ + β∂rγ˜θθ + 2βrγ˜θθ − σ23γ˜θθB , (3.86)where B is the divergence of the shift vector,B(t, r) = Diβi = ∂rβ +2βr+ β(∂r γ˜rr2γ˜rr+∂r γ˜θθγ˜θθ). (3.87)To display the equation of motion for the trace of the extrinsic curvature K and A˜ijwe first defineDij ≡ DiDjα , (3.88)which has 2 independent components,Drr = ∂2rα− ∂rα(∂rγ˜rrγ˜rr+ 4∂rφ), (3.89)1223.6. BSSN in Spherical SymmetryDθθ = r∂rαγ˜θθγ˜rr+r22∂rα(∂rγ˜θθγ˜rr+ 4∂rφγ˜θθγ˜rr). (3.90)The trace of Dij isD ≡ γijDij = e−4φ(Drrγ˜rr+ 2Dθθr2γ˜θθ). (3.91)Then the evolution of K is given by∂tK = −D + α(13K2 +A˜2rrγ˜2rr+ 2A˜2θθγ˜2θθ)+ β∂rK + 4πα(ρ+ S) (3.92)and the evolution equations for the traceless part of the extrinsic curvature are∂tA˜rr = e−4φ[−DTFrr + α (RTFrr + 8πSTFrr )]+ α(A˜rrK − 2A˜2rrγ˜rr)+ 2A˜rr∂rβ + β∂rA˜rr − σ23BA˜rr , (3.93)∂tA˜θθ =e−4φr2[−DTFθθ + α (RTFθθ + 8πSTFθθ )]+ α(A˜θθK − 2A˜2θθγ˜θθ)+ 2βrA˜θθ + β∂rA˜θθ − σ23A˜θθB , (3.94)1233.6. BSSN in Spherical Symmetrywhere R denotes the 3-Ricci tensor with non-vanishing componentsRrr =3(∂r γ˜rr)24γ˜2rr− (∂r γ˜θθ)22γ˜2θθ+ γ˜rr∂rΛ˜ +12∂rγ˜rrΛ˜+1r(4∂rφ− ∂rγ˜rr − 2∂r γ˜θθγ˜θθ− 2γ˜rr∂rγ˜θθγ˜2θθ)− 4∂2rφ+ 2∂rφ(∂rγ˜rrγ˜rr− ∂rγ˜θθγ˜θθ)− ∂2r γ˜rr2γ˜rr+2(γ˜rr − γ˜θθ)r2γ˜θθ, (3.95)Rθθ =r2γ˜θθγ˜rr(∂rφ∂rγ˜rrγ˜rr− 2∂2rφ− 4(∂rφ)2)+r2γ˜rr((∂r γ˜θθ)22γ˜θθ− 3∂rφ∂r γ˜θθ − 12∂2r γ˜θθ)+ r(Λγ˜θθ − ∂r γ˜θθγ˜θθ− 6∂rφγ˜θθγ˜rr)+γ˜θθγ˜rr− 1 . (3.96)In the above expressions the superscript TF denotes application of the trace-free-part operator, whose action can be written explicitly asXTFrr = Xrr −13γrrX =23(Xrr − AXθθBr2), (3.97)XTFθθ = Xθθ −13γθθX =13(Xθθ − BXrrA). (3.98)Here X represents any of the tensors D , R or S.1243.7. Scalar Field Synamics and Energy-Momentum Tensor in Spherical SymmetryFinally, the evolution of Λ˜i reduces to∂tΛ˜ = β∂rΛ˜− ∂rβΛ˜ + 2αγ˜rr(6A˜θθ∂rφγ˜rr− 8πSr − 23∂rK)+αγ˜rr(∂rγ˜rrA˜rrγ˜2rr− 2∂rγ˜θθA˜θθγ˜2θθ+ 4A˜θθγ˜rr − γ˜θθrγ˜2θθ)+ σ(23Λ˜B +∂rB3γ˜rr)+2rγ˜θθ(∂rβ − βr)− 2∂rαA˜rrγ˜2rr. (3.99)3.7 Scalar Field Synamics and Energy-MomentumTensor in Spherical SymmetryHere we present the evolution equations of a complex scalar field, with an arbitrarypotential V , minimally coupled to gravity. The governing equations for a masslessreal scalar field follow as a special case where the potential and the imaginary partof the field are both set to zero.The geometry of spacetime is given by a generic metric in spherical symmetry:ds2 = (−α2 + β2a2)dt2 + 2a2βdtdr + a2dr2 + r2b2dΩ2, (3.100)where a, b, α and β are all functions of t and r and where a and b are related to theprimary BSSN variables via a = γ˜rr exp(4φ) and b = γ˜θθ exp(4φ).The complex scalar field is given in terms of real and imaginary parts, ΨR andΨI , respectively,Ψ = ΨR(t, r) + iΨI(t, r) , (3.101)1253.7. Scalar Field Synamics and Energy-Momentum Tensor in Spherical Symmetryand has an associated energy-momentum tensorTµν = ∇µ∇νΨR − 12gµν∇ηΨR∇ηΨR+ ∇µ∇νΨI − 12gµν∇ηΨI∇ηΨI− 12gµνV (|Ψ|) . (3.102)The evolution of the real part of the scalar field can be reduced to a pair of first-order-in-time equations via the definitionΞR ≡ b2aα(∂tΨR − β∂rΨR) . (3.103)We then find the following evolution equations for ΨR and ΞR:∂tΨR =αb2aΞR + β∂rΨR , (3.104)∂tΞR =αb2a(∂2rΨR + 2∂rΨRr)+ ∂rΨR∂r(αb2a)+ β∂rΞR + ΞR∂rβ + ΞR2βr− aαb2∂2|Ψ|V (|Ψ|) . (3.105)The evolution equations for ΨI and ΞI follow from the index substitutions R ↔ Iin the right hand sides of (3.104) and (3.105), respectively.The matter source terms in the G-BSSN equations, namely ρ, S, Si, Sij, can besimplified by defining Π and Φ asΠ ≡ aα(∂tΨ− β∂rΨ) ≡ ΠR(t, r) + iΠI(t, r) , (3.106)ΠR =aα(∂tΨR − β∂rΨR) = ΞRb2, (3.107)1263.7. Scalar Field Synamics and Energy-Momentum Tensor in Spherical SymmetryΠI =aα(∂tΨI − β∂rΨI) = ΞIb2, (3.108)Φ ≡ ∂rΨ ≡ ΦR(t, r) + iΦI(t, r) , (3.109)ΦR = ∂rΨR , (3.110)ΦI = ∂rΨI . (3.111)Using these definitions, the variables ρ and S are given byρ(t, r) =|Π|2 + |Φ|22a2+V (|Ψ|)2, (3.112)S(t, r) =3|Π|2 − |Φ|22a2− 32V (|Ψ|) . (3.113)In spherical symmetry, Si has only a radial component,Si = [Sr(t, r), 0, 0] , (3.114)withSr = −ΠRΦR +ΠIΦIa. (3.115)Similarly, the spatial stress tensor, Sij , has only two independent components,Sij =Srr(t, r) 0 00 r2Sθθ 00 0 r2 sin2 θSθθ , (3.116)1273.7. Scalar Field Synamics and Energy-Momentum Tensor in Spherical SymmetrywithSrr =|Π|2 + |Φ|22− a2V (|Ψ|)2, (3.117)Sθθ = b2( |Π|2 − |Φ|22a2− V (|Ψ|)2). (3.118)128Chapter 4Non-linear Gravity WaveEvolutions with the G-BSSNFormulation4.1 IntroductionAs first discussed in the introduction, the General Relativistic theory of gravity hasa radiative component to it, whereby the waves in the metric of pure vacuum carryinformation and energy. In Cartesian coordinates the linearized Einstein’s equation(1.13), i.e. perturbation near the flat spacetime, has a simple planar wave solution(choosing z to be the direction of propagation):h¯ij = Hijei(kz±ωt) , (4.1)where Hij can be written as a linear combination of two basis tensors, e+ and e×:Hij = ae+ij + be×ij . (4.2)Here e+ is a tensor with only nonzero components: e+xx = −e+yy = 1 and e× is atensors for which the nonzero components are: e×xy = e×yx = 1.The solutions to the linearized tensorial wave equation in spherical-polar coor-dinates with axial symmetry are known as Teukolsky waves [96]. Another vacuum1294.1. Introductionaxisymmetric ansatz to Einstein’s equations was proposed by Brill [97]—which canbe considered both in linear and non-linear regime. Both of these solutions arecommonly used in gravitational waves evolutions, and we will introduce them in ourdiscussion of the non-linear regime in Sec. 4.2.4.The very first study of pure vacuum solutions of Einstein’s equation using Brilldata is due to Eppley [98] who demonstrated that a sufficiently strong pure vacuumBrill configuration contains a black hole. The first dynamical study of pure vacuumusing a numerical approach was [99] and also adopted the Brill initial data (in axialsymmetry). This study was the first simulation where pure vacuum dynamicallycollapses to a black hole. The first axisymmetric numerical evolutions of Teukolskywaves was due to Abrahams and Evans [100] who showed the formation of a blackhole from an imploding Teukolsky wave packet.The first full 3-dimensional simulations of gravity waves was by Shibata andNakamura [31], who were the first to propose the rescaling of the BSSN formulationand perform successful long term evolutions of small amplitude waves. Follow upwork by Baumgarte and Shapiro [32] completed the BSSN formulation by introduc-ing the conformal connection functions and showed the much improved performanceof BSSN formulation in comparison to the free ADM evolutions of small amplitudegravitational waves.The 3D near-linear regime was first explored in [101, 102] with reports of nu-merical difficulties in achieving long-time dynamics. Strong gravity dynamics of 3Dpure vacuum Brill data was first performed in [103] and [104] with the primary goalof finding black hole critical solutions. The evolution of Brill initial data in axialsymmetry was revisited later in [105], again in search of critical behaviour. Howeveras we discuss in the following, the critical phenomena in gravitational waves collapseis as yet an unsolved problem.Critical Phenomena in Pure Gravity Waves CollapseThe follow up work to [100] by Abrahams and Evans [106] is the first report on1304.2. Equations of Motion for Strong Gravity Waves Dynamicsthe observation of type II critical behaviour in the collapse of Teukolsky waves. Inaddition, in [107], they found evidence for universality of the solution with a mass-scaling exponent γ ≈ 0.38, surprisingly close to Choptuik’s finding for the masslessscalar field, and an echoing exponent ∆ ≈ 0.5. However, there are 5 studies [77–80, 83] which further investigate the universality of the solution, all of which reportunsuccessful attempts to reproduce the original results. Among these, Sorkin’s work[80] stands out. He finds evidence for scaling in agreement with γ ≈ 0.38 and hintsof a DSS structure. However, the measured echoing exponent differs from [106](∆ ≈ 1.1) for his choice of Brill initial data. In addition, Sorkin reports observationof a “ring of singularity” forming in the near-critical regime, which is somewhatunexpected and peculiar. These inconsistencies motivate a revisit of the problem,with a new axisymmetric code which implements the G-BSSN formulation and thatappeared to be promising based on the spherical case we studied in the previouschapter.In the rest of this chapter we discuss the development of a G-BSSN axisymmet-ric code and its application to non-linear gravity waves dynamics. The numericaltechniques and the implementation of a new G-BSSN-based code in cylindrical coor-dinate is presented in Sec. 4.3 as is the initialization process using two types of Brilland Teukolsky-type initial data. In Sec. 4.4 we present the primary calculationsperformed using the code to evolve the pure vacuum in the strong gravity regime.Discussions of the future steps that are required to optimize and bring the code toproduction for type II critical phenomena studies are given in Equations of Motion for Strong Gravity WavesDynamicsIn this section, we summarize the G-BSSN equations and present their form in axialsymmetry using cylindrical coordinates. As mentioned in the previous chapter, G-1314.2. Equations of Motion for Strong Gravity Waves DynamicsBSSN is a generalization of the BSSN formulation to curvilinear coordinates wherethe flat 3-metric is not the unity matrix (see the discussion in Sec. 3.2.1). Forexample, in cylindrical coordinates, the flat 3-metric is given by:ds2 = dρ2 + dz2 + ρ2dϕ2 , (4.3)which we denote as γ˚ij and has determinant γ˚ = ρ2. Here, we will not derive the G-BSSN equations in any detail, but rather refer the reader to [88] for a full derivation(also see the discussion in the previous chapter). In summary, then, the G-BSSNequations in vacuum are given by:∂tφ = −16αK + βi∂iφ+ σ16D˜kβk , (4.4)∂tγ˜ij = −2αA˜ij + βk∂kγ˜ij + γ˜ik∂jβk + γ˜kj∂iβk − σ23A˜ijD˜kβk , (4.5)∂tK = −γijDjDiα+ α(A˜ijA˜ij + 13K2) + βi∂iK , (4.6)∂tA˜ij = e−4φ (−DiDjα+ αRij)TF + α(KA˜ij − 2A˜ilA˜lj)+ βk∂kA˜ij + A˜ik∂jβk + A˜kj∂iβk − σ23A˜ijD˜kβk , (4.7)∂tΛ˜i = ∂tΓ˜i − Γ˚ijk∂tγ˜jk , (4.8)in which we have set the matter sources to zero and written out the Lie derivativesexplicitly. Here Γ˚ijk denotes the Christoffel symbols associated with flat metric γ˚ij .As in the previous chapter, σ is a parameter that determines the two standardchoices of G-BSSN for the evolution of the determinant of the 3-conformal-metric,1324.2. Equations of Motion for Strong Gravity Waves Dynamicsγ˜:σ = 1 ⇒ ∂tγ˜ = 0 , (Lagrangian choice) , (4.9)σ = 0 ⇒ (∂t −L~β)γ˜ = 0 , (Lorentzian choice) . (4.10)All of the simulations in this chapter use the Lagrangian option.In (4.8) the first term, ∂tΓ˜i, is given by:∂tΓ˜i = −2A˜ij∂jα+ 2α(Γ˜ijkA˜kj − 23γ˜ij∂jK + 6A˜ij∂jφ)+ γ˜lj∂j∂lβi+ βj∂jΓ˜i − Γ˜j∂jβi + σ3[2Γ˜iD˜kβk + γ˜li∂l(D˜kβk)], (4.11)and in the second term, Γ˚ijk∂tγ˜jk, the time evolution of the inverse of the confor-mal metric, ∂tγ˜jk, can be evaluated using the time evolution of γ˜ij , (4.5) by theconsideration:γ˜ikγ˜kj = δij ⇒ ∂t(γ˜ikγ˜kj) = 0⇒ (∂tγ˜ik)γ˜kj + γ˜ik∂tγ˜kj = 0⇒ (∂tγ˜ik)γ˜kj γ˜jl + γ˜ik(∂tγ˜kj)γ˜jl = 0⇒ ∂t(γ˜ik)δkl + γ˜ik(∂tγ˜kj)γ˜jl = 0⇒ ∂t(γ˜il) = −γ˜ik(∂tγ˜kj)γ˜jl , (4.12)in which ∂tγ˜kj should be replaced by the right hand side (RHS) of (4.5). In addition,in (4.11), Γ˜i is substituted by the re-defined conformal connection, Λ˜i, in G-BSSNvia its definition (3.12):Λ˜k ≡ γ˜ij(Γ˜kij − Γ˚kij)≡ Γ˜k − Γ˚kij γ˜ij ⇒ Γ˜k = Λ˜k + γ˜ijΓ˚kij . (4.13)We remind the reader that the 3-Ricci tensor, Rij, in (4.7) is computed using the1334.2. Equations of Motion for Strong Gravity Waves DynamicsG-BSSN primary variables as:Rij = Rφij + R˜ij , (4.14)in which:Rφij = −2D˜iD˜jφ− 2γ˜ijD˜kD˜kφ+ 4D˜iφD˜jφ− 4γ˜ijD˜kφD˜kφ , (4.15)R˜ij = −12γ˜lm∂m∂lγ˜ij + γ˜k(i∂j)Γ˜k + Γ˜kΓ˜(ij)k + γ˜lm(2Γ˜kl(iΓ˜j)km + Γ˜kimΓ˜klj). (4.16)In the expression for the conformal Ricci tensor, R˜ij, we again use (4.13) to eliminateΓ˜k since in G-BSSN, Λ˜k is the primary dynamical variable, rather than Γ˜k.The Hamiltonian and momentum constraints for pure vacuum are given by:H ≡ γ˜ijD˜iD˜jeφ − eφ8R˜+e5φ8A˜ijA˜ij − e5φ12K2 = 0 , (4.17)Mi ≡ D˜j(e6φA˜ji)− 23e6φD˜iK = 0 , (4.18)and are only used at the initial time in the G-BSSN formulation. As we will discuss,we monitor the constraints as a diagnostic tool.4.2.1 G-BSSN in Cylindrical Coordinate with Axial SymmetryWe now proceed by imposing the axial symmetry in cylindrical coordinates. As-suming that ∂ϕ is a Killing vector, we consider the following form for the conformal3-metric in cylindrical coordinates:γ˜ij =γ˜ρρ(t, ρ, z) γ˜ρz(t, ρ, z) 0γ˜ρz(t, ρ, z) γ˜zz(t, ρ, z) 00 0 ρ2γ˜ϕϕ(t, ρ, z) . (4.19)1344.2. Equations of Motion for Strong Gravity Waves DynamicsThis reduces to the flat 3-metric (4.3) when the off-diagonal component, γ˜ρz vanishesand the diagonal terms are unity. The traceless part of the extrinsic curvature hassimilar non-zero components:A˜ij =A˜ρρ(t, ρ, z) A˜ρz(t, ρ, z) 0A˜ρz(t, ρ, z) A˜zz(t, ρ, z) 00 0 ρ2A˜ϕϕ(t, ρ, z) , (4.20)and all of the other terms on the right hand side of the G-BSSN equations (4.4,4.8)such as Rij , DiDjα and the Lie derivative terms, have similar non-zero components,consistent with these ansatzes.Furthermore, due to the axial symmetry, the shift vector and conformal connec-tion function can only have non-zero components in the ρ and z directions:βi = [βρ(t, ρ, z), βz(t, ρ, z), 0] , (4.21)Λ˜i = [Λ˜ρ(t, ρ, z), Λ˜z(t, ρ, z), 0] . (4.22)4.2.2 Coordinate ChoicesFor this study, we use the standard 1+log and Gamma-driver35 conditions to evolvethe lapse and shift36,∂tα(t, ρ, z) = −2αK , (4.23)∂tβi = µΛ˜i − ηβi , (4.24)35The choice of β = 0, which worked fine in the spherical case in the previous chapter, is notan effective coordinate choice here. In particular, using β = 0 the off-diagonal component of theconformal 3-metric grows and causes the metric to become singular (non-invertable).36We note that the code implements an advective version of these equations with the βi∂i termon the LHS and a more general expression on the RHS, aαK + bα2K, for experimental purposes.However the simulations shown in the results section use only the standard choice.1354.2. Equations of Motion for Strong Gravity Waves Dynamicswhere µ and η are adjustable parameters. For most of the calculations presentedhere we use µ = 3/4 and η ≈ 1/(10M), where M is the total mass of the system(ADM mass). As will be discussed in Sec. 4.3.2, we choose an initialization suchthat the mass is of order 1, therefore η ≈ 1/(10M) will be of order 10− Note on Complexity and Regularity of the EquationsWe note that the G-BSSN equations (4.4,4.8), even limited to axial symmetry, con-tain tens of thousands of terms if the right hand side of the equations are expressedin terms of the conformal metric γ˜ij and the rest of the primary variables of G-BSSN. To reduce this complexity, and avoid potentially repetitive calculations, weintroduce the inverse of the conformal metric γ˜ij as a new set of “work” variables inthe numerical solver. Further, we compute the various covariant derivatives usingsome of the components of the conformal Christoffel symbols defined as additionalwork variables:Γ˜ρρρ(t, ρ, z) =12γ˜ρρ∂ργ˜ρρ + γ˜ρz∂ργ˜ρz − 12γ˜ρz∂z γ˜ρρ , (4.25)Γ˜ρzρ(t, ρ, z) =12γ˜ρz∂ργ˜ρρ + γ˜zz∂ργ˜ρz − 12γ˜zz∂z γ˜ρρ , (4.26)Γ˜ρzz(t, ρ, z) = γ˜ρρ∂z γ˜ρz − 12γ˜ρρ∂ργ˜zz +12γ˜ρz∂z γ˜zz , (4.27)Γ˜zρρ(t, ρ, z) =12γ˜ρz∂ργ˜ρρ + γ˜zz∂ργ˜ρz − 12γ˜zz∂z γ˜ρρ , (4.28)Γ˜zzρ(t, ρ, z) =12γ˜ρz∂z γ˜ρρ +12γ˜zz∂ργ˜zz , (4.29)Γ˜zzz(t, ρ, z) = γ˜ρz∂z γ˜ρz − 12γ˜ρz∂ργ˜zz +12γ˜zz∂z γ˜zz , (4.30)Γ˜ϕϕz(t, ρ, z) =12∂z γ˜ϕϕγ˜ϕϕ. (4.31)Specifically, we use the left hand side (LHS) symbols in in the G-BSSN equations,while their values are given by the RHS and stored in separate work variables. For1364.2. Equations of Motion for Strong Gravity Waves Dynamicsthe remaining components of the conformal Christoffel symbols:Γ˜ϕϕρ(t, ρ, z) =1ρ+12∂ργ˜ϕϕγ˜ϕϕ, (4.32)Γ˜zϕϕ(t, ρ, z) = −ργ˜ρz γ˜ϕϕ −12ρ2γ˜ρz∂ργ˜ϕϕ − 12ρ2γ˜zz∂z γ˜ϕϕ , (4.33)Γ˜ρϕϕ(t, ρ, z) = −ργ˜ρργ˜ϕϕ −12ρ2γ˜ρρ∂ργ˜ϕϕ − 12ρ2γ˜ρz∂z γ˜ϕϕ , (4.34)we explicitly use the RHS expressions in the G-BSSN equations. This is important,as they contain powers of ρ and similarly our ansatz for the conformal metric (4.19)has explicit ρ2 dependency and its inverse has explicit 1/ρ2 term. Therefore, atseveral places in the symbolic calculations, the powers of ρ appear and can canceland simplify to regular terms. An example of such a cancellation is in the termγ˜lm(2Γ˜kl(iΓ˜j)km + Γ˜kimΓ˜klj)in the Ricci tensor (4.16). In addition, Γ˜ϕϕρ contains asingular 1/ρ term, which will be appropriately eliminated by the subtraction of theflat background Christoffel symbols (˚Γϕϕρ = 1/ρ) in the re-definition of the conformalconnection (4.13).Even with the use of the Christoffel symbols as intermediate variables, the LHSof G-BSSN equations are rather lengthy and of course need to be derived using atensor manipulation software. For example , the ρρ component of the 3-Ricci tensor1374.2. Equations of Motion for Strong Gravity Waves Dynamicsthat appears in the RHS of the ρρ component of Eq. (4.7) is given by:Rρρ = R˜ρρ +Rφρρ =− 12γ˜ρρ∂2ρ γ˜ρρ −12γ˜zz∂2z γ˜ρρ − γ˜ρz∂z∂ργ˜ρρ + γ˜ρz∂ρΛ˜z + γ˜ρρ∂ρΛ˜ρ+12Λ˜z∂z γ˜ρρ +12Λ˜ρ∂ργ˜ρρ + ∂z γ˜ρρ(32γ˜ρzΓ˜ρρρ +32γ˜zzΓ˜ρzρ +12γ˜ρzΓ˜zzρ +12γ˜ρρΓ˜zρρ)+ ∂ργ˜ρρ(32γ˜ρρΓ˜ρρρ +32γ˜ρzΓ˜ρzρ)+ ∂ργ˜zz(−12γ˜zzΓ˜zzρ −12γ˜ρzΓ˜zρρ)+ ∂ργ˜ρz(γ˜ρρΓ˜zρρ + γ˜ρzΓ˜zzρ)+ ∂z γ˜ρz(2γ˜ρzΓ˜zρρ + 2γ˜zzΓ˜zzρ)+14∂ργ˜2ϕϕγ˜2ϕϕ+1ρ(−12∂ργ˜ρργ˜ϕϕ− ∂ργ˜ϕϕγ˜ϕϕ+ γ˜ρρ∂ργ˜ϕϕγ˜2ϕϕ)+1ρ2(γ˜ρργ˜ϕϕ− 1)⋆+ ∂ρφ(2Γ˜ρρρ + 2γ˜ρργ˜ρρΓ˜ρρρ + 4γ˜ρργ˜ρzΓ˜ρzρ + 2γ˜ρργ˜zzΓ˜ρzz)+ ∂zφ(2Γ˜zρρ + 2γ˜ρργ˜ρρΓ˜zρρ + 4γ˜ρργ˜ρzΓ˜zzρ + 2γ˜ρργ˜zzΓ˜zzz)+ ∂ρφ(−γ˜ρργ˜ρρ ∂ργ˜ϕϕγ˜ϕϕ− γ˜ρz γ˜ρρ∂z γ˜ϕϕγ˜ϕϕ)+ ∂zφ(−γ˜ρργ˜ρz ∂ργ˜ϕϕγ˜ϕϕ− γ˜ρργ˜zz ∂z γ˜ϕϕγ˜ϕϕ)+ (∂ρφ)2 (4− 4γ˜ρργ˜ρρ)− 4(∂zφ)2γ˜ρργ˜zz + 4γ˜ρργ˜ρz∂ρ∂zφ− 2∂2ρφ− 2γ˜ρργ˜ρρ∂2ρφ− 2γ˜ρργ˜zz∂2zφ+1ρ(−2γ˜ρργ˜ρρ∂ρφ− 2γ˜ρργ˜ρz∂zφ) ⋆ , (4.35)in which we highlighted the potentially irregular terms containing negative powersof ρ. We note that these terms are indeed regular in the limit ρ → 0 as we willdiscuss shortly, but it is nonetheless crucial to collect the powers of ρ and computethese terms in an appropriate way to prevent round-off errors. For instance, if theterm containing 1/ρ2 is computed as two separate terms: γ˜ρρ/(ρ2γ˜ϕϕ) and −1/ρ2,both these terms are large (in fact diverging) floating point numbers at the vicinityof the origin, and if added to any other regular term, before being summed together,can create a significant round-off error. However, if the subtraction γ˜ρρ/γ˜ϕϕ − 1 isperformed first, the value is a small floating point number, (behaving as ρ2 in thelimit ρ→ 0 as we will discuss in Sec. 4.3.4) and the division by ρ2 creates a number1384.2. Equations of Motion for Strong Gravity Waves Dynamicsthat is of the same order as the rest of the terms in the expression.4.2.4 Axisymmetric Initial DataThe initialization of a pure gravity wave is done by specifying a non-trivial form forthe conformal 3-metric, γ˜ij , and then solving the Hamiltonian constraint (4.17) forthe conformal factor eφ. The momentum constraint (4.18) is satisfied trivially byrequiring that the initial data be time symmetric:A˜ij = K = 0 . (4.36)Furthermore, without loss of generality, one can assume that at the initial time theρ and z coordinates are orthogonal, i.e. the off-diagonal term in (4.19) is zero:γ˜ij(t = 0, ρ, z) =A(ρ, z) 0 00 B(ρ, z) 00 0 ρ2C(ρ, z) , (4.37)in which the functions A, B and C can be chosen arbitrarily. In addition it is easyto perform a coordinate transformation z → z′ (at initial time slice) to make thezz and ρρ components of γ˜ij equal (any two-metric can be written in a conformallyflat form). Therefore, a suitably generic axisymmetric initial conformal 3-metric isgiven by:γ˜ij(t = 0, ρ, z) =eV (ρ,z) 0 00 eV (ρ,z) 00 0 ρ2W (ρ, z) . (4.38)Here we use eV for the first two diagonal components of the conformal metric to beconsistent with the literature. Note that this form for γ˜ij holds only at the initialtime; during the evolution the metric evolves to a form given by (4.19). In this1394.2. Equations of Motion for Strong Gravity Waves Dynamicschapter, we particularly focus on two different types of initial data:W (ρ, z) = 1 , (4.39)andW (ρ, z) = e−2V (ρ,z) . (4.40)4.2.5 Brill Initial DataThe first choice (4.39), known as Brill initial data, yields the conformal metric:γ˜(B)ij (t = 0, ρ, z) ≡eV (ρ,z) 0 00 eV (ρ,z) 00 0 ρ2 , (4.41)for which the Hamiltonian constraint simplifies to:0 = H =(∂2ρ + ∂2z +1ρ∂ρ)ψ(ρ, z) +18ψ(ρ, z)(∂2ρ + ∂2z)V (ρ, z) . (4.42)Here ψ = eφ is the conformal factor, V (ρ, z) will be chosen to a localized function,with proper behaviour in the limit ρ→ Teukolsky-type Initial DataThe second option (4.40) creates a conformal 3-metric with the following form:γ˜(T )ij (t = 0, ρ, z) ≡eV (ρ,z) 0 00 eV (ρ,z) 00 0 ρ2e−2V (ρ,z) , (4.43)1404.2. Equations of Motion for Strong Gravity Waves Dynamicsand the Hamiltonian constraint simplifies to:0 = H =(∂2ρ + ∂2z +1ρ∂ρ − ∂ρV ∂ρ − ∂zV ∂z)ψ(ρ, z)− 18ψ(ρ, z)[(∂2ρ + ∂2z +4ρ∂ρ)V (ρ, z)− 2(∂ρV )2 − 2(∂zV )2].(4.44)We refer to this initialization as Teukolsky-type initial data as it somewhat mimicsthe Teukolsky wave [96] in the weak field limit, V (ρ, z) ≪ 1. However, Teukolsky’ssolution is not time symmetric, while our choose (4.36) is time symmetric.The main difference between the Brill initialization (4.41) and Teukolsky-typedata is the fact that the Teukolsky-type initial data has determinant equal to that ofthe flat cylindrical metric, while that is not the case for Brill data. In the weak fieldlimit, V (ρ, z) << 1, this translates to a difference in the trace of the two linearizedmetrics. The Teukolsky-type initial metric can be written as:γ˜(T )ij ≈ γ˚ij + η(T )ij , (4.45)in which γ˚ij is the flat cylindrical metric, and η(T )ij is the linearized deviation and istraceless:η(T )ij =V (ρ, z) 0 00 V (ρ, z) 00 0 −2ρ2V (ρ, z)⇒ Tr(η(T )ij ) = γ˚ijη(T )ij = 0 , (4.46)which parallels the traceless property of the linearized solution to the Einstein’sequations in cylindrical coordinate. This solution, as will be seen, propagates simi-larly to a wave packet. However, in a similar linearization of the Brill initial data,1414.2. Equations of Motion for Strong Gravity Waves Dynamicsthe perturbation metric, η(B)ij , is not traceless:η(B)ij =V (ρ, z) 0 00 V (ρ, z) 00 0 0⇒ Tr(η(B)ij ) = γ˚ijη(B)ij = 2V (ρ, z) , (4.47)and as we will show in Sec. 4.4, the Brill data does not propagate like a wave packet,even in the weak field limit.4.2.7 Computing the ADM Mass of the Gravitational PulseThe ADM mass[15] is defined as:MADM ≡ 116π∫∂Σ∞√γγjnγim (∂jγmn − ∂mγjn) dSi , (4.48)where ∂Σ∞ denotes the boundary surface at spatial infinity, dSi = Ni√γ|∂Σ∞dX2,is the surface element vector, dX2 = dX1dX2, where X1 and X2 are the coordinateson the boundary surface, γ|∂Σ∞ is the determinant of the reduced metric on ∂Σ∞,and finally, Ni is the unit normal vector to ∂Σ∞ (NiNi = 1). As described in[28], for a metric that is asymptotically conformally flat, (4.48) can be written as avolume integral:MADM = − 12π∫Σdx3√γ˜D˜2ψ . (4.49)Further, substituting D˜2ψ from the Hamiltonian constraint we have,MADM =116π∫Σdx3√γ˜(ψ−7A˜ijA˜ij − ψR˜− 23ψ5K2). (4.50)Finally, since A˜ij = K = 0 at the initial time the ADM mass simplifies to:MADM = − 116π∫Σdx3√γ˜ψR˜ , (4.51)1424.3. Numericswhich is a convenient expression for computing the ADMmass using BSSN variables.4.3 NumericsIn this section, we summarize the numerical techniques used to initialize and evolvethe G-BSSN system of equations. We also describe the diagnostic tools we imple-mented to validate the results and test the correctness of the numerical solver aswell as the equations themselves.4.3.1 Numerical GridBefore discretizing the PDEs of the G-BSSN formulation, we first transform to non-uniform spatial coordinates denoted by ρ˜ and z˜ (their definition will follow shortly).This coordinate transformation has two purposes: 1) to compactify the coordinateby mapping the domain of the cylindrical coordinate to a numerical domain thatcontains spatial infinity:D(ρ,z) = [0,+∞]× [−∞,+∞]→ [ρ˜min, ρ˜max]× [z˜min, z˜max] = D(ρ˜,z˜) , (4.52)2) increasing resolution toward the origin, (ρ, z) = (0, 0). Choosing a non-lineartransformation, a uniform mesh in the compactified coordinate (ρ˜, z˜) is effectively anon-uniform grid in the cylindrical coordinate. The mesh step sizes in each directionof the coordinates are related by (approximately):∆ρ =∂ρ(ρ˜)∂ρ˜∆ρ˜ , (4.53)∆z =∂z(z˜)∂z˜∆z˜ , (4.54)Therefore, by defining the functions ρ˜(ρ) and z˜(z) such that their derivatives de-crease toward the origin we can achieve the desired non-uniform grid. Specifically,1434.3. Numericswe choose the following two functions:ρ = exp(ρ˜)− exp(δ) + 11− ρ˜/R∞ −11− δ/R∞ , (4.55)z =12(exp(z˜ + δ)− exp(−z˜ + δ) + 11− (z˜ + δ)/R∞ −11− (−z˜ + δ)/R∞).(4.56)The reader may easily check that the cylindrical coordinate domain, [0,+∞]ρ ×[−∞,+∞]z, is mapped to the numerical domain [δ,R∞]ρ˜× [−(R∞−δ),+(R∞−δ)]z˜ .We also define the notations ∂Σ˜:∂Σ˜ ≡ ∂Σ˜zρ=∞ ∪ ∂Σ˜z=−∞ρ ∪ ∂Σ˜z=+∞ρ (4.57)in which:∂Σ˜zρ=∞ ≡ [R∞]ρ˜ × [−(R∞ − δ),+(R∞ − δ)]z˜ (4.58)∂Σ˜z=−∞ρ ≡ [δ,R∞]ρ˜ × [−(R∞ − δ)]z˜ (4.59)∂Σ˜z=+∞ρ ≡ [δ,R∞]ρ˜ × [+(R∞ − δ)]z˜ (4.60)to denote the outer numerical boundaries that correspond to asymptotically flatspatial infinity. The inner boundary, which is the symmetric axis, is simply denotedby (ρ = 0, z) and corresponds to (ρ˜ = −δ, z˜) on the numerical grid.The parameters R∞ and δ can be adjusted to change the numerical location ofspatial infinity and the origin, therefore effectively changing the distribution of gridpoints across the spatial domain. For the choice (4.55,4.56) a uniform grid on thecompactified coordinates creates an approximately uniform grid on a logarithmicscale, ln(ρ) and ln(z), at the vicinity of the origin. Such behaviour is presented inFig. 4.1 in a rather coarse grid with 64 points (to better demonstrate the distribution1444.3. Numerics−4 −2 0 2 4−4−3−2−101234ln(x)x˜  x = z, x˜ = z˜ − δx = ρ, x˜ = ρ˜x = ρ, x˜ = ρ˜lFigure 4.1: Distribution of grid points on a non-uniform grid with 64 points forthe choice of (4.55,4.56) for ρ and z coordinates (green and blue curve respectively)using the parameters: δ = −4 and R∞ = 4. Points are distributed uniformly acrossthe compactified coordinate (vertical axis, x˜). As it is clear from both curves, thepoints are also approximately distributed uniformly in the logarithm of the spatialcoordinate (horizontal axis lnx) for a range that expands up to ln(x) ≈ −3. Notethat the blue curve that corresponds to the z coordinate is shifted by constantδ to have the same range as ρ˜. For comparison, we also plotted a uniform gridstructure (the red curve), in which the 64 points are distributed uniformly in a linearcoordinate ρ = aρ˜l + b that maps (−4, 4) to the range ρ ∈ (0, 54) (to match the lastpoints of the other two graphs: ln(54) ≈ 4). As is clear, the choices (4.55,4.56)provides much more resolution in the vicinity of the origin, x = 0, compared to thered curve. Approximately half of the grid points are in the region ln(x) < 0 for ourlogarithmic coordinate choice, while a uniform grid has only 1 grid point next tothe origin in that region, while the rest of the points are located in a region that isnot of interest when the gravitational wave is focused in the vicinity of the origin.1454.3. Numericsof the points).4.3.2 InitializationWe use three different types of profile to initialize the function V in (4.39,4.40) thatdefines the conformal metric. The standard choice of a localized function is often aGaussian profile:V (ρ, z) = Aρ2 exp(−(ρ− ρ0)2∆2ρ− (z − z0)2∆2z). (4.61)We also use the following seed functions (choice of F is similar to [107]) to createwave packets:F (r) := Aκλ5[1−( rλ)2]6, (4.62)G(r) := Aκλ(1− ( rλ)2)4. (4.63)These seed functions are used as following to initialize V :V (ρ, z) =(F (4)(r − r0)r− 2F(3)(r − r0)r2)sin2 θ , (4.64)V (ρ, z) =(G(1)(r − r0)r− 2G(r − r0)r2)sin2 θ . (4.65)where F (n) denotes the n-th derivative of F . Specifically, we use a Gaussian profileand G for Brill-type initial data, and F and G for a Teukolsky-type initial wavepacket. Here κ is a normalization factor (different for the two functions) chosensuch that when the amplitude A is set to a value of order 1—along with the typicalchoices of λ in this study—the ADM mass of the gravity wave pulse is also of order1. In particular we have chosen κ ≈ 3.7× 10−4 for F , and for G, κ is approximately1. The parameter λ determines the typical length of the wave packet. r is the radial1464.3. Numerics0 5 10 15ργ˜ρρ(t=0,ρ,z=0)=eV=γ˜zz(0,ρ,0)  V = V (F ), HighV = V (F ), LowV = V (G), HighV = V (G), LowFigure 4.2: Initial profile of the conformal metric component γ˜zz or γ˜ρρ along theradial direction (θ = π/2, or z = 0 ) in cylindrical coordinates for two choices of Fand G in (4.64,4.65). Here the solid lines are the strong initial data that collapsesand are associated with amplitudes A = 3.0 and A = 2.0 in Eq. (4.65) and Eq. (4.64)respectively. The dashed lines are still in a non-linear regime but will disperse afterreflecting back from the center. They are associated with amplitudes A = 1.0 andA = 0.6 in (4.65) and (4.64) respectively. For the seed function F the rest of theparameters are set to: r0 = 15 and λ = 5.0. For G as a seed function, the parametersare: r0 = 10 and λ = 3. As one can see, F creates a wave packet with 5 extremawhile G has a simpler form with 2 extrema.1474.3. Numericsdistance from the center:r2 ≡ ρ2 + z2 , (4.66)and sin(θ) is the polar angle defined as:sin θ ≡ ρr. (4.67)The typical shapes of the initial conformal metric constructed from the seed func-tions F and G are given in Fig. 4.2 for strong (black hole formation) and weak(dispersal) initial data.The lapse function is set to unity at the initial time:α(t = 0, ρ, z) = 1 , (4.68)and since we choose time symmetric initial data, we have:βρ(t = 0, ρ, z) = βz(t = 0, ρ, z) = 0 , (4.69)A˜ij(t = 0, ρ, z) = K(t = 0, ρ, z) = 0 , (4.70)at the initial time. The conformal connection function, Λ˜k, is initialized using itsdefinition and the given initial conformal metric:Λ˜k(t = 0, ρ, z) = γ˜ij(t = 0, ρ, z)(Γ˜kij(t = 0, ρ, z) − Γ˚kij). (4.71)Finally, the G-BSSN variable φ is initialized from its relation to the conformal factor:φ(t = 0, ρ, z) = ln(ψ(ρ, z))|H(ψ)=0 (4.72)in which |H(ψ)=0 denotes that ψ solves the Hamiltonian constraint, (4.42) or (4.44)1484.3. Numericsdepending on the choice of Brill or Teukolsky-type initialization. In both case, ψsatisfies the boundary conditions:∂ρψ(ρ, z)|(ρ=0,z) = 0 (4.73)ψ|∂Σ˜ = 1 (4.74)where ∂Σ˜ is defined in (4.57), and denotes the 3 outer boundaries of the numericaldomain corresponding to spatial infinity.We note that the multigrid solver that we use (PAMR’s default MG solver) issensitive to imposing inner boundary conditions such as (4.73). Therefore we usea regularized version of the Hamiltonian constraint equations (4.42,4.44) where weuse L’Hospital’s rule to replace the irregular term on the axis by:1ρ∂ρψ|ρ→0 = ∂2ψ∂ρ2(4.75)In addition, to evaluate the finite difference equivalent of the term ∂2ρψ on theaxis, we use the fact that ψ is an even function in ρ due to the symmetry. Thedetailed description of dealing with such boundaries and the ghost-cell-equivalentimplementation of boundary conditions in our finite differencing toolkit (FD) isdescribed in the Appendix.4.3.3 Boundary ConditionsAxial symmetry demands that the diagonal metric components be even functionsin ρ while the off-diagonal term is an odd function:∂ργ˜ρρ(t, ρ, z)|(ρ=0,z) = ∂ργ˜zz(t, ρ, z)|(ρ=0,z) = ∂ργ˜ϕϕ(t, ρ, z)|(ρ=0,z) = 0 , (4.76)1494.3. Numericsγ˜ρz(t, ρ = 0, z) = 0 . (4.77)The trace-free extrinsic curvature obeys analogous conditions to the conformal 3-metric:∂ρA˜ρρ(t, ρ, z)|(ρ=0,z) = ∂ρA˜zz(t, ρ, z)|(ρ=0,z) = ∂ρA˜ϕϕ(t, ρ, z)|(ρ=0,z) = 0 , (4.78)A˜ρz(t, ρ = 0, z) = 0 . (4.79)The extrinsic curvature and φ are scalars, and are therefore even functions in ρ(invariant under ρ→ −ρ symmetry):∂ρK(t, ρ, z)|(ρ=0,z) = ∂ρφ(t, ρ, z)|(ρ=0,z) = 0 . (4.80)The reflection of the ρ components of the conformal connection vector, Λ˜ρ, and theshift vector, βρ, under ρ→ −ρ symmetry result in a negative sign. Therefore theyare odd functions in ρ:Λ˜ρ(t, ρ = 0, z) = βρ(t, ρ = 0, z) = 0 . (4.81)Their z components, on the other hand, remain unchanged under ρ→ −ρ symmetry,therefore they are even functions:∂ρΛ˜z(t, ρ, z)|(ρ=0,z) = ∂ρβz(t, ρ, z)|(ρ=0,z) = 0 . (4.82)Forward (one-sided stencil) finite difference approximations of these boundary con-ditions are used to update the value of the functions on the axis.Note that for the odd functions described above, beside the value of the function,1504.3. Numericsthe second derivative along ρ also vanishes on the axis:∂2ρΛ˜ρ(t, ρ, z)|(ρ=0,z) = ∂2ρβρ(t, ρ, z)|(ρ=0,z) = 0 . (4.83)∂2ρ γ˜ρz(t, ρ, z)|(ρ=0,z) = ∂2ρA˜ρz(t, ρ, z)|(ρ=0,z) = 0 . (4.84)We impose a forward (one-sided) finite difference equivalent of these conditionsto update values at the point next to the axis to improve the smoothness of thefunctions.At infinity, the boundary conditions are rather simple, since we are working ina compactified coordinate, and are given by the asymptotically flat values:γ˜ij(t, ρ, z) =1 0 00 1 00 0 1 at : (ρ, z) ∈ ∂Σ˜ , (4.85)A˜ij = K = φ = βi = Λ˜i = 0 at : (ρ, z) ∈ ∂Σ˜ , (4.86)α = 1 at : (ρ, z) ∈ ∂Σ˜ . (4.87)4.3.4 Evolution SchemeTo evolve the G-BSSN system we implemented a Crank-Nicholson implicit scheme.As discussed in the previous chapter, this implicit scheme appears to be essential todeal with the coordinate singularity ρ = 0. There is an extra boundary conditionthat is implicit in the system (due to the symmetry)—the ρ→ 0 limiting behaviorof the two components:γ˜ρρ = C +O(ρ2) , (4.88)1514.3. Numericsγ˜ϕϕ = C +O(ρ2) , (4.89)where the constant C is the same for both. Equivalently, their difference shouldbehave as:γ˜ρρ − γ˜ϕϕ = O(ρ2) , (4.90)and a violation of this condition will manifest itself, for example, in a diverging termin the Ricci component previously shown in (4.35):Rρρ =1γ˜ϕϕ(γ˜ρρ − γ˜ϕϕρ2)+ · · · (4.91)Our implementation of a implicit scheme appears to be sufficient to keep the systemof equations regular on the axis with no need to explicitly regularize the equationsat the analytic level. However, similar to the spherical case of the previous chapter,Kreiss-Oliger dissipation is crucial to suppress numerical noise which can particularlyaffect the near-origin evolution.4.3.5 Note on G-BSSN’s Additional ConstraintsWe emphasize that we were only able to achieve stable evolution using the so calledLagrangian choice (4.10) for the evolution of the determinant of the conformal met-ric. In this choice, the determinant of the conformal metric is given by:γ˜ = ρ2(γ˜ρργ˜zz − γ˜2ρz)γ˜ϕϕ =ρ2 for Teukolsky-typeρ2e2V (ρ,z) for Brill data(4.92)and is equal to its initial time value as given above for both the Teukolsky-type andBrill data. This constraint can be monitored or can be imposed to fix one of thecomponents of the γ˜ij metric. Our numerical experiments suggest that imposingthis equation improves the performance of the code. In particular, we have applied1524.3. Numericsthis constraint to compute the γ˜zz component of the 3-conformal-metric. Similarly,the tensor A˜ij is expected to remain traceless during the evolution—which we havemonitored as an accuracy gauge. However, this equation can also be imposed toimprove the accuracy, and we use it to fix the same component as the componentfixed in 3-conformal metric, i.e. A˜zz.4.3.6 Tests: Convergence of Primary VariablesAs discussed in previous chapters, the first diagnostic test is to perform a conver-gence test. This test validates the consistency of the finite difference approximationand determines whether the finite difference numerical solution is converging to anunderlying continuum function well-resolved in the discretized mesh. The conver-gence factor,Q(t; q) =||qh(t,X) − qh/2(t,X)||2||qh/2(t,X)− qh/4(t,X)||2, (4.93)has been plotted for three of the primary variables of the G-BSSN equations,g˜rr,K, Λ˜z and the RHS of the evolution equation for A˜ρz for a strongly gravitatingevolution in Fig. 4.3. As can be seen from this plot, all variables exhibit at leastfirst order convergence and some second order convergence—values of log2(Q) be-tween 2 and 1. We particularly chose to measure the convergence factor for theRHS of ∂tA˜ρz, since it is one of the computationally most complex expressions todiscretize and evaluate, and contains several “irregular”, 1/ρ and 1/ρ2 terms, whichcan potentially be sources of further numerical error. This function appears to havethe smallest convergence factor, yet it is at least first order convergent and at bestsecond order convergent as demonstrated in the figure. Note that the convergencecurves are plotted for three resolutions: 128× 256, 256× 512 and 512× 1024 whichare not particularly high resolutions and one might expect to observe better conver-gence at higher resolutions. We note that we performed a similar test for all of theother primary G-BSSN variables, and they are all about second order convergent.1534.3. NumericsFigure 4.3: Convergence factor for the G-BSSN variables: g˜rr,K and Λ˜z. The con-vergence factor (4.93) for these functions is plotted during the evolution of a strongpure gravity wave. As presented in the plot, they all converge in a second orderto first order fashion corresponding to the values 2 and 1 in log2(Q) respectively.The last curve is the convergence factor for the RHS of the evolution equation forA˜ρz which is a rather complicated function of 3-metric and the rest of the G-BSSNvariables and contains several “irregular” terms (containing 1/ρ and 1/ρ2). Again,we observe at least first order convergence for this function.1544.3. Numerics4.3.7 Tests: Conservation of Constraints During EvolutionWe also monitor the momentum constraint (4.18), which has only ρ and z compo-nents, and the Hamiltonian constraint (4.17). Since the G-BSSN formulation is afree evolution system, these constraints are not imposed during the evolution of thegeometry and can be used as an effective diagnostic tool to gauge the accuracy ofthe numerical time integration of the equations.Fig 4.4(a) and Fig 4.4(b) demonstrate the second order convergence to zero of thetwo components of the momentum constraint as the resolution improves. Results forthe Hamiltonian constraint are presented in Fig 4.4(c) for 4 consecutively decreasingresolutions and similarly show second order convergence to zero. Furthermore, sincethe evaluator of the Hamiltonian constraint in this plot uses a different form of theequation and a different finite differencing scheme, the convergence of the its valueat t = 0 is also an independent test for the multi-grid initial value solver that wehave used to solve the Hamiltonian constraint at the initial time.Beside the Hamiltonian and momentum constraints, here we also present anotherconstraint that naturally arises from the definition of the conformal connection Λ˜i(4.13). Since Λ˜i is evolved via a time dependent equation (4.8), the differencebetween its evolved value and its computed value, Λ˜kc from the conformal metric,Λ˜kc = γ˜ij(Γ˜kij − Γ˚kij), (4.94)forms a residual: Λ˜i− Λ˜ic, that should converge to zero as h→ 0. Fig 4.4(d) presentsthe value of the ρ component of this residual, ||Λ˜ρ − Λ˜ρc ||l2, in a log2 scale, wherea factor of 2 decrease for the resolution refinement h → h/2 signals a second orderconvergence to zero. Such behaviour is clearly present in the plot.Note that we also monitored all of the other constraints that occur in the G-BSSN formulation, including: the trace of A˜ij , which by definition should remainzero and the determinant of the conformal 3-metric, γ˜, which should stay equal to1554.3. NumericsFigure 4.4: Evolution of the conserved variables: (a): The time evolution of thenorm of the ρ component of the momentum constraint (4.18), Mρ, for 4 differentresolutions h, h/2, h/4, h/8, plotted in a log2 scale. Each step of resolution improve-ment by a factor of 2 results in the decrease of the value of the function by a factorof about 4, (2 in the scale of the plot) which demonstrates second order convergenceto zero. (b): Similar to (a) for the z component of the momentum constraint. (c):The norm of the Hamiltonian constraint (4.17) at 4 different resolutions similar to(a). Again, we observe second order convergence to zero. The convergence of thevalue of the curve at t = 0 is an independent test for the validity of the initial valuesolver implemented to solve the Hamiltonian constraint. (d): The norm of the ρcomponent of the residual, Λ˜i− Λ˜ic which exhibits second order convergence to zeroas the resolution improves. This suggests that the free evolution of the conformalconnection, Λ˜i via (4.8) in G-BSSN formulation is consistent with its definition in(4.13).1564.3. Numericsits value at initial time for the Lagrangian choice (4.10). Of course, these tests areonly meaningful if we do not enforce these constraints during the test runs. Asmentioned before, after the test runs, we do impose these constraints during theevolution of the results presented below.4.3.8 Tests: Direct Validation via Einstein’s EquationsFinally, the most robust test we developed involves a direct evaluation of the com-ponents of the Einstein’s equations for a given 4-metric, gµν , reconstructed from the3-metric, γij , and the coordinate variables:gµν = −α2 + βlβl βiβj γij (4.95)where the 3-metric and coordinate functions are given by the solutions the G-BSSNsolver produces. We define the residual for the Einstein’s equation:Eµν ≡ Gµν − 8πT µν . (4.96)For the case of pure gravity waves, since the energy-momentum and the Ricci scalarare zero37, the residual can be defined equal to the Ricci tensor:Eµν ≡ Rµν , (4.97)and is zero if and only if the Einstein’s equation is solved correctly in vacuum.The norm of various components of the residual (4.97) are plotted in Fig. 4.5,and the plots suggest that the residual converges to zero in a second order fashion,providing strong evidence that the computed metric does indeed satisfy the Ein-stein’s equations. We also note that this is not only a validation of the correctness37Taking the trace of the Einstein’s equation: Rµν −12Rgµν = 8πTµν we get: −R = 8πT whereT is the trace of the energy-momentum tensor. Therefore, in vacuum, Tµν = 0, the Ricci scalar iszero.1574.3. NumericsFigure 4.5: Convergence of the Einstein equations residuals defined in (4.97) fora strong gravity evolution. The curves present 4 different consecutively improving(by factor of 2) mesh sizes, and various components of the residual are plotted here.As it is clear on the log2 scale of the graph, the value of the residual decreasesby a factor of about 4 at each step of resolution refinement. This convergence tozero in second order fashion suggests that the numerical solver that provides the4-metric to the independent residual evaluator code is indeed computing a metricthat satisfies the Einstein’s equation. This direct validation further suggests thatthe set of equations (G-BSSN) used in the numerical solver are equivalent to theEinstein’s equations.1584.4. Resultsof our numerical solver, but also proves that the derivation of G-BSSN equations wasdone correctly, and that the equations are equivalent to Einstein’s equations. Wenote that we developed the testing facilities independently of the numerical solver.In particular, we derived the residual (4.97) using a different tensor manipulationpackage than the one used to derive the G-BSSN equations. We also used a differentfinite difference scheme than the one adopted to discretize the G-BSSN evolutionequations.4.4 ResultsIn this section we provide examples of the performance of the code for evolvingpure gravity initial data that is set to be in the non-linear regime, slightly above orbelow the threshold of black hole formation. We note that the code is developedfor a generic axisymmetric system with matter content and can be used with othernumerical solvers to evolve the coupled matter-gravity systems.4.4.1 Evolution of Teukolsky-type Initial DataFirst we present a typical evolution of the Teukolsky-type initial data using the seedfunction F (4.62,4.64) that creates the initial wave packet plotted in Fig. 4.2. Thezz component of the conformal metric γ˜ij is plotted in Fig. 4.6. The amplitude isset to about 10% less than the critical value. Even though the waves eventuallycompletely disperse, this is still a strong field evolution. As can be seen from theplots, the system exhibits an intermediate confined state where the gravitationalwave packet is held together by its own gravity. The system at the initial time isonly slightly away from the linear regime—the amplitude of the conformal metricdiffers from 1 by only ≈ 0.05—however as the wave focuses toward the origin, theself-gravitation amplifies and the system exhibits non-linear dynamics.An evolution of a similar configuration but where the amplitude is now about10% higher than the critical value is presented in Fig. 4.7. Here, the wave packet1594.4. Resultsgets trapped by its own self gravitation and collapses to form a black hole. Asdemonstrated in the last snapshots of the evolution in Fig. 4.7 the conformal metricshows large deviations from the flat metric. The wave packet is confined in thevicinity of the origin where the lapse function starts to collapse as a black holeforms. Comparing Fig. 4.7 and Fig. 4.6 one can observe the two distinct end statesof the evolution of the system: black hole and dispersal.Another initialization that we implemented uses the seed function G, whichcreates a wave packet with 2 extrema and a simpler overall structure. Fig. 4.8 and4.9 demonstrate the two distinct end state of the evolution, dispersal and black holeformation. Again, the dispersal data is set to a non-linear regime and is close to thecritical value, therefore as the wave packet experiences a non-linear evolution nearthe origin, the reflected wave develops a different wave front structure as shown inthe last snapshot in Fig. 4.8.Finally, in Fig. 4.10 we plot the central value of the lapse function for the collaps-ing and dispersing Teukolsky-type wave shown in Fig. 4.8 and 4.9. The two distinctend states are apparent from the fact that the dispersal data has an intermediatetime where the lapse function decreases to small value, while for the large initialamplitude, the lapse collapses and a black hole forms.4.4.2 Evolution of Brill Initial DataIn this section, we demonstrate some of the simulations that we performed usingBrill initial data. As discussed, one important difference between Teukolsky-typeand Brill initial data is in the trace of the conformal metric in the linearized regimewhere a Teukolsky-type wave packet satisfies the traceless condition, while the Brilldata does not. We have experimented with the Gaussian initialization (4.61), thatis commonly used for Brill initial data in the literature, but for comparison herewe demonstrate the simulations of Brill initial data that are initialized using a seedfunction G similar to Fig. 4.8-4.9.1604.4. ResultsFigure 4.6: Evolution of a non-linear Teukolsky-type wave packet: Here the grav-itational wave is initialized by the seed function F , and as can be seen from thesnapshot t = 0, has 5 extrema. All of the snapshots plot the γ˜zz component of theconformal metric. The initial data is time symmetric, therefore part of the wavepacket moves inward while the other part disperses toward infinity as seen in ththee t = 9.72 snapshot. During the intermediate time, 12.96 < t < 21.60, the wavepacket forms a gravitational geon confined to the vicinity of the origin. Eventually,the wave starts to disperse to infinity at t > 23.76. The system is in a strong gravityregime and as can be seen from the last snapshot at t = 32.40 the non-linear effectschanges the outgoing shape of the wave packet relative to t = 0. This calculation,as well as the other simulations presented in this chapter, are performed on a non-uniform grid with resolution 512 × 1024. Here, we have transformed the functionsback to the spatial (ρ, z) coordinates for demonstration purposes.1614.4. ResultsFigure 4.7: Collapse of Teukolsky-type wave initial data: Here the amplitude ofthe initial pulse is large enough to cause the system to collapse. The wave packet isinitialized using the seed function F and similar to Fig. 4.6 we are plotting the γ˜zzcomponent of the conformal metric. As one can observe, in the later snapshots, thepulses accumulate and become confined within a very small region close to the originand eventually system collapses to a black hole. The simulations take about 100hours using 32 CPUs. The non-uniform structure of the numerical grid is observablein the first snapshot.1624.4. ResultsFigure 4.8: Non-linear evolution of a Teukolsky-type wave packet: Various snap-shots of the zz component of the conformal metric for a Teukolsky-type data ini-tialized using the seed function G (4.63) are shown. Compared to Fig. 4.6, the wavepacket has a simpler structure. Again, the time symmetric initial data has both aningoing and an outgoing part (seen at t = 4.80 snapshot). After an intermediatenon-linear phase where the metric components deviate strongly from 1, the wavepacket disperses as seen in t = 27.60, where the outgoing form of the wave is some-what different than the initial time. The central value of the lapse function for thisevolution is shown in Fig. 4.10.1634.4. ResultsFigure 4.9: Typical evolution of collapsing Teukolsky-type data using the seedfunction G. Here, similar to Fig. 4.7, we the γ˜zz component of the conformal metric.The wave packet focuses toward the origin and eventually the system collapses toa black hole. The collapse of the lapse function (singularity avoidance property) atthe vicinity of the origin slows down the evolution. The central value of the lapsefunction for this evolution is plotted in Fig. 4.10.1644.4. ResultsFigure 4.10: Time evolution of central lapse: The logarithm of the lapse functionat the origin α(t, ρ = 0, z = 0) is plotted versus the time coordinate. Here “high”denotes the collapse scenario and “low” is the dispersal case that are illustratedin Fig. 4.9 and 4.8 respectively. As one can see, the central value of the lapse fordispersal data exhibits an intermediate time with significantly large central red-shift,but eventually rises back to 1 as the wave packet leaves the origin. However, in acollapse scenario, the central lapse collapses, as shown by the solid curve.1654.4. ResultsFigure 4.11: Dispersal evolution of Brill initial data: Plotted is the γ˜zz componentof the conformal metric at different times during a dispersal Brill data initializedusing the seed function G, (4.63,4.65). Note the large difference between the initalamplitude here and the initial amplitude of Teukolsky data in Fig. 4.8. As is appar-ent in the second snapshot (t = 6.93) the Brill “wave” has pure gauge content, thatdoes not propagate. The ingoing part of the wave in the second snapshot (t = 6.93)is located at ρ ≈ 2.5 while the outgoing part is at ρ = 13, however in between thereis a third pulse at ρ = 8 (same radius as the initial data) that does not propagate.The ingoing wave eventually reflects back from the center and in snapshots t = 20.79and t = 23.73 starts to move outward. Note that in most of the diagrams the gaugecontent is clearly observable and is located at about ρ ≈ 8 without moving. In fact,in the last snapshot the reflected wave has passed through the gauge pulse and islocated at ρ ≈ 15.1664.4. ResultsFigure 4.12: Collapse of Brill data: Evolution of strong initial data that eventuallycollapse to a black hole. Again, similar to Fig. 4.11, the wave packet contains a puregauge component, which does not propagate. The ingoing pulse eventually focusesto a very small region and collapses to a black hole. The central value of the lapsefor this evolution is given in Fig. 4.13.1674.4. ResultsFigure 4.13: Central lapse for the Brill data evolution: α(t, ρ = 0, z = 0) as afunction of time in a logarithmic scale. Similar to the Teukolsky-type data (Fig. 4.10)the dispersal case exhibits an intermediate high red-shift while eventually rising backto the unity, while during a collapse process the lapse function at the center collapsesand slows down the evolution which is expected from the singularity avoidanceproperty of the 1+log slicing.1684.5. Further Remarks and ConclusionThe dispersal scenario is shown in Fig. 4.11 where the initial wave packet has asimpler shape to Teukolsky-type evolution shown in Fig. 4.8. However, first note thelarge difference in the amplitude in the first snapshots of the two figures. Secondly,as described in the caption of Fig. 4.11 the wave packet has a pure gauge contentthat simply remains at a fixed point as the radiative part of the wave moves inwardand outward. This evolution eventually disperses and the gravitational radiationpropagates toward infinity. However, the pure gauge content remains in the nu-merical domain and the final state of the conformal metric is not unity. A collapsescenario is plotted in Fig. 4.12. As one can see, similar to the dispersal case, thewave packet contains a pure gauge component that remains at a fixed radius. Thein-going part of the time symmetric data eventually collapses and a black hole forms.For both dispersal and collapse cases, the central value of the lapse is plottedin Fig. 4.13. As seen from the dashed curve, the dispersal data experiences a largecentral red-shift while eventually the wave packet disperses. However, the collapse isdistinct and the lapse function collapses as the black hole forms. One can comparethis to Fig. Further Remarks and ConclusionIn this chapter, we described a new G-BSSN axisymmetric code. The code is im-plemented such that it can be coupled to arbitrary matter content. However, wemeasured the performance of the code in the most challenging scenario: pure non-linear gravity waves evolution. We demonstrated both Teukolsky-type and Brillinitial data evolution and discussed their difference. In particular, the Teukolsky-type wave packet appears to be a better choice since it mimics the linearized regimewave-type propagation better than Brill initial data. Our results suggest that G-BSSN is a promising formulation to evolve pure gravity waves and further extensionof this work can shed more insight into the expected type II critical solution in puregravity waves.1694.5. Further Remarks and ConclusionHowever, there are several other steps required to bring the code to production.Specifically, resolution higher than 512 × 1024 is needed to capture discrete selfsimilarity. This can be achieved by further optimizing the code, as well as by usinglarger number of CPUs. At the moment, our numerical Hamiltonian constraintsolver for the initialization is rather slow and has limited our work to this resolution.In addition, the number of grid functions can be decreased by carefully examiningthe dependencies of fields which in turn will reduce the CPU communication timeand improve the scaling of the code to higher number of CPUs and higher resolution.170Chapter 5ConclusionIn this thesis we presented three projects in critical phenomena studies in gravi-tational collapse. Chapter 2 focused on the Einstein-Vlasov system and the ob-servation of type I behaviour and weak universality in the system. Chapter 3demonstrated a possibility for using G-BSSN formulation in type II critical phe-nomena studies where, for the first time, we presented an implementation of a fullyevolutionary system that can successfully derive the spacetime evolution close tothe critical regime and find a type II threshold solution. 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RNPL Reference Manual,http://laplace.physics.ubc.ca/People/matt/Rnpl/index.html, (1995).183Appendix AAppendix: FD, FiniteDifference ToolkitA.1 IntroductionFD is a set of Maple [109, 110] routines and definitions designed to handle varioustasks in applying finite difference techniques in solving partial differential equations(PDEs). Particularly, it is developed to provide a methodology and a syntacticlanguage to solve time dependent or boundary value PDEs arising in physics. Solvinga PDE involves various complications, including finding the correct finite differenceapproximation (FDA) to a specific accuracy, dealing with boundary points on thediscretized numerical domain, initialization, developing testing facilities for insuringaccuracy, and finally creating routines to solve the FDA equations over the numericaldomain. FD is designed to simplify these steps while providing full control over theentire process, allowing the user to focus on the underlying physical phenomena.Specifically, FD is not created to be a “blackbox” PDE solver, rather it provides amixed level of automation and user controlled definitions.FD is still under development and was originally designed to be used in thenumerical relativity research where the computational task to numerically solve theEinstein’s equations 38, is rather challenging. Keeping that in mind, FD was devel-oped to deal with PDEs and differential expressions that are lengthy (in some case38A set of 10 highly complex and non-linear coupled PDEs that govern the dynamics of the curvedspacetime in strongly gravitating objects like black holes or neutron stars.184A.1. Introductionthousands or tens of thousands of expressions) and are usually machine generated toavoid human error. Therefore, FD is written in the Maple language, which providesa powerful symbolic manipulation environment and unifies the process of derivingthe continuum form of the PDEs, and applying finite difference methods to createa discretized form. Furthermore, FD is built to directly parse a given differentialexpression39 in its canonical continuous form 40 in Maple. This eliminates the needfor having another high-level specification to define a PDE which can be a cumber-some task for the user, especially if the PDEs are derived from tensorial equations –such as PDEs arising in general relativity. This prevents potential human errors intransferring the equations from the symbolic calculation environment to the target“PDE solver” environment. In addition, FD inherits all of the capabilities of Maplelanguage to deal with PDEs and algebraic expressions. In particular, the user canmanage their working environment using Maple’s built-in data and control struc-tures and use PDEtools package to implement various other tasks such as coordinatetransformation and checking for the consistency of the equations.41After posing a PDE as a set of FDA equations over a discretized domain, theseequations can be solved using FD’s default point-wise Newton-Gauss-Sidel relax-ation algorithm (see Sec. A.2.2) – which is a common method in solving nonlineartime dependent PDEs. FD generates Fortran subroutines (and C wrappers) toperform the relaxation and may be used as a rapid prototyping tool to implementvarious finite difference schemes to solve a PDE. It also provides a rapid developmentwork-flow to create routines to evaluate the residual of the given FDA expression asa diagnostic tool.FD is capable of dealing with the boundaries of the numerical domain by provid-ing a syntax to specify the PDE or boundary conditions differently at different parts39PDE, written in the from: D(f) = 0, where D is a differential operator and f is the unknownfunction, would be a special case of a differential expression that is equal to zero.40An expression in which derivatives are presented using Maple’s diff operator. An example ofsuch expression is: diff(f(t,x),t,t) - diff(f(t,x),x,x)41We note that GRTensor [111] Maple package is available for dealing with tensorial partialdifferential equations and tensor manipulation.185A.1. Introductionof the discretized domain. This allows the user to impose various boundary condi-tions such periodic boundary conditions, asymptotic behaviour boundary conditionsor inner boundary conditions. This, particularly, is achieved in FD by implementingan equivalent method to the ghost cell technique used in finite difference methods,and can be used to create inner boundary conditions that arise from the symmetriesin the system – such as requesting particular functions to be even or odd in specificcoordinate direction.In FD’s environment, specifying the finite difference scheme by the user is assimple as merely providing the order of accuracy and limitation on the allowedgrid points in the Finite Difference Molecule (FDM). FD has a simple internalalgorithm to determine the number of points required to do “forward”, “backward”and “centered” finite differencing of a given partial differential expression with thegiven accuracy. It ensures that the generated stencil expression has accuracy that isequal to the user specified value or better. The computed stencils are all stored inan internal table and are user accessible to be monitored for their order of accuracyand form.Finally, FD produces Fortran routines (and C wrappers) that are parallel-readyand can be used in the framework of a high performance computing infrastructure.This is achieved by passing boundary flags to the routines which specify if theboundaries of the grid are between CPUs or are real physical boundaries. FD adoptsPAMR’s [33] standard in this matter, but any other parallelization framework shouldalso have a similar method to deal with the inner CPU boundaries. We note that theFortran routines generated by FD use only the basic data types of Fortran languageand creating wrappers to communicate with them from a different language shouldbe a straightforward task. By default, FD generates the C language wrappers whichis one of the most common languages in high performance computing.This user manual describes all of the features mentioned above and introducesthe syntax of FD for posing a PDE as a finite difference equation with the given186A.2. Overview of Finite Difference Methodboundary conditions. First, two algebraic types are defined which are the fundamen-tal objects that FD uses to identify a finite difference expression. These types arethe building blocks that FD uses to directly translate a PDE to a discretized equa-tion and eventually to Fortran routines. Then, a derived Maple table is introducedthat specifies the PDE and the boundary conditions over the discretized numericaldomain. Finally, we present the utilities FD provides to choose a finite differencescheme, compute the FDA equivalent of a given PDE and create Fortran codes tosolve it. We assume that the reader has a working knowledge of Maple programmingand is familiar with the basic concepts of finite difference methods. Some of theseconcepts are reviewed in Sec. A.2. An experienced user may skip this section, whilethose who are not are encouraged to consult the references [109, 110, 112, 113].A.2 Overview of Finite Difference MethodFinite difference methods are numerical techniques to express continuum differen-tial expressions/equations as (approximate) algebraic expressions/equations. Theresulting expression is known as the Finite Difference Approximation (FDA). AnFDA for a derivative term, such as df(x)/dx, at a given point x, is a combinationof the values of the function at certain points in the vicinity of x. For instance,values at the points {f(x), f(x+∆x), f(x+2∆x)} (discretized values) can be usedto approximate the first derivative of the function as:df(x)dx≈ −3f(x) + 4f(x+∆x)− f(x+ 2∆x)2∆x, (A.1)where ∆x is the step size of the discretization. This “scheme” is called forward finitedifferencing, as the discrete values are extended in positive(forward) x direction.Similarly, one can use the points {f(x), f(x − ∆x), f(x + ∆x)} to compute the187A.2. Overview of Finite Difference Methodsecond derivative of the function,d2f(x)dx2≈ f(x−∆x)− 2f(x) + f(x+∆x)∆x2. (A.2)Here the point x is at the center, and thus the scheme is named centered finite differ-encing. The discretized points, (· · · , x−∆x, x, x+∆x, · · · ), construct a domain foran Ordinary Differential Equation (ODE) or a Partial Differential Equation (PDE).The following diagram illustrates this concept of discretized numerical domain for a1+1 (1 spatial, 1 time) dimensional spacetime:✉ ✉ ✉✉ ✉ ✉✉ ✉ ✉xixi−1 xi+1fnifni−1 fni+1fn−1ifn+1itn−1tntn+1A discretization method transforms a function from a continuum form to a dis-crete form symbolized as:f(t, x)→ f(tn, xi) ≡ fni . (A.3)Here, we denote the time indexing with the superscript n and the spatial indexingusing the subscript symbols (i, j, k). The grid structure, ∪xi × ∪tn, (and similarlyin higher dimensions add yj and zk) is usually considered to be uniform:tn = t0 + n∆t ≡ t0 + nht , (A.4)188A.2. Overview of Finite Difference Methodxi = xmin + i∆x ≡ xmin + ihx , (A.5)yj = ymin + j∆y ≡ ymin + jhy . (A.6)Using these symbols, a partial differential expression such as ∂xf(t, x) can be writtenas:∂f(t, x)∂x=f(t, x+ hx)− f(t, x− hx)2hx+O(h2x) =fni+1 − fni−12hx+O(h2x) , (A.7)and the wording “approximation” is due to the neglecting of the O(h2x) term. Herethe function O(h2x) has explicit dependency of the from h2x on the step size, andrepresents the error of the approximation (or equivalently can be interpreted as the“accuracy” of the FDA). Replacing all of the derivatives with FDA expressions, aPDE becomes an algebraic equation for the discrete values of the function. Forexample, consider performing the following FDA on the heat equation,∂f(t, x)∂t+ α∂2f(t, x)∂x2= 0 → fn+1i − fniht+ αfni+1 − 2fni + fni−1h2x= 0 , (A.8)where in the discretized version of the equation, the unknown is the vector:Fn+1 = fn+1i , (A.9)and is to be solved numerically for a given Fn. Obviously knowing the values F1,i.e the initial time profile of the function f , the process of solving Fn+1 in terms ofFn means, by induction, finding the entire solution on the time domain indexed byn.189A.2. Overview of Finite Difference MethodA.2.1 Computing the FDA ExpressionThere is a systematic method to find the FDA of the l’th derivative of a function,dlf(x)/dxl. Consider L points, in the vicinity of x as:{x+ q1∆x, x+ q2∆x, · · · , x+ qL∆x} , (A.10)where qi’s are L distinct integers usually chosen in a minimalistic fashion such thatx+ qi∆x is close to x. For example, the forward and centered finite differencing inEq. (A.1) and Eq. (A.2) are associated with:{q1, q2, q3}forward = {0, 1, 2} , {q1, q2, q3}center = {−1, 0, 1} . (A.11)Using these L points, and L unknown coefficients {β1, β2, β3, · · · , βL} one can createL Taylor expansions upto truncation error O(∆xL),β1f(x+q1∆x) = β1F(0)+β1q1F(1)+β1q21F(2)+· · ·+β1ql1F (l)+· · ·+β1q(L−1)1 F (L−1) ,(A.12)β2f(x+q2∆x) = β2F(0)+β2q2F(1)+β2q22F(2)+· · ·+β2ql2F (l)+· · ·+β2q(L−1)2 F (L−1) ,(A.13)...βLf(x+qL∆x) = βLF(0)+βLqLF(1)+βLq2LF(2)+· · ·+βLqlLF (l)+· · ·+βLq(L−1)L F (L−1) ,(A.14)where we defined:F (r) =drf(x)dxr(∆x)rr!, (A.15)190A.2. Overview of Finite Difference Methodand F (0) = f(x). Then we can find the coefficients {β1, β2, β3, · · · , βL} such thatsumming over the entire right hand sides of the equations, all of the F (r) terms havecoefficients zero, except F (l) which can be set to have coefficient 1. This processleads to the following set of L linear equations for βi’s:L∑m=1βmf(x+ qm∆x) = F(0)∑mβm + F(1)∑mqmβm + F(2)∑mq2mβm+ · · ·+ F (l)∑mqlmβm + · · ·+ F (L−1)∑mq(L−1)m βm = F(l)⇒∑mβm = 0∑mq1mβm = 0∑mq2mβm = 0...∑mql−1m βm = 0∑mqlmβm = 1∑mql+1m βm = 0...∑mq(L−1)m βm = 0For L distinct given qi’s, this linear system has a unique solution vector which wedenote by β⋆i . Note that the left hand side of the summation is a finite difference191A.2. Overview of Finite Difference Methodexpression:L∑m=1β⋆mf(x+qm∆x) = F(l) =dlf(x)dxl(∆x)ll!⇒ dlf(x)dxl=l!(∆x)lL∑m=1β⋆mf(x+qm∆x) ,(A.16)and therefore we find the desired FDA expression for the l’th derivative using Lneighboring points. In this calculation, clearly one should assume,L ≥ l + 1 , (A.17)which simply indicates that finding the FDA of a l’th derivative term requires at leastl+1 points. The truncation error in the Taylor expansions is O(∆xL) and since thefinite difference sum is divided by ∆xl in Eq. (A.16) the accuracy of the final finitedifference expression is at least O(∆x(L−l)). However in certain cases (for examplein centered scheme) the finite difference expression can have higher accuracy as thecoefficient in the next leading O(∆xL) term in the summation happens to simplifyto zero. The reader may verify this for the FDA given in Eq. (A.2)This calculation is internally performed by FD as it encounters derivative termsin a PDE and returns the FDA equivalent of them.42 There is a simple front-endfunction (mostly for demonstration purposes) in FD:Sten(diffexpr,[points])which calls the internal FDA operator on the given differential expression, diffexpr,and computes the stencil using the points, [points], (denoted by {qi} in the sys-tematic derivation above). For example in the following we demonstrate the com-putation of the forward and centered FDA in Eq. (A.1) and Eq. (A.2) for the firstand second derivatives respectively:42We note that FD does not use any of Maple’s substitution/replacement procedures, rather itperforms recursively to parse a PDE and return FDA equivalents of its differential expressions.192A.2. Overview of Finite Difference Method> Sten(diff(f(x),x),[0,1,2]);-3 f(x) + 4 f(x + h) - f(x + 2 h)1/2 ---------------------------------h> Sten(diff(f(x),x,x),[-1,0,1,2,3]);11 f(x - h) - 20 f(x) + 4 f(x + 2 h) + 6 f(x + h) - f(x + 3 h)1/12 --------------------------------------------------------------2hExample 1: Simple FDA of derivatives using FDWe emphasize that this procedure is solely for demonstration purposes, and acts onlyon a single derivative term. In practice, FD uses a different procedure, Gen Sten,that performs the FDA operation according to an FDA scheme specification providedby the user, and it performs on arbitrary length PDEs.A.2.2 Iterative Schemes for Non-Linear PDEsSolving a time dependent PDE for a function f(t, ~x) involves integrating the equationforward in time, given the initial value f(0, ~x). In the discrete language of finitedifferencing, this process reduces to finding the advanced time level value of thefunction, fn+1ijk , for the given current value, fnijk. Starting with the “initial data”,f0ijk, the time integration can be performed by applying this process consecutivelyfor Nt time levels:Initial Data fn=0i,j,k → fn=1i,j,k → . . . → fn=Nti,j,k Final State (A.18)To demonstrate this update process, let’s revisit the 1-D heat equation, with adifferent discretization scheme (known as leap-frog):∂f(t, x)∂t+ α∂2f(t, x)∂x2= 0 → fn+1i − fn−1i2ht+ αfni+1 − 2fni + fni−1h2x= 0 . (A.19)193A.2. Overview of Finite Difference MethodThis finite difference equation (FDE), is a second order approximation to the PDEat the point (tn, xi), and it involves values of the function at that point, and thepoints in the vicinity of it. The FDE includes the following points:{(n+ 1, i), (n − 1, i), (n, i − 1), (n, i), (n, i − 1)} . (A.20)and the“unknown” in this set, as highlighted in (A.19), is fn+1i . This set of pointsis called the Finite Difference Molecule (FDM) and is illustrated in the followingdiagram for the FDA of heat equation (A.19):✈ ✈ ✈✈ ✈ ✈✈ ✈ ✈xixi−1 xi+1fnifni−1 fni+1fn−1ifn+1itn−1tntn+1FDM depends on the finite difference scheme. For example, consider a different(also second order accurate) FDA of the heat equation at the point (tn+1/2, xi),where tn+1/2 denotes the point tn + ht/2:∂f(t, x)∂t+α∂2f(t, x)∂x2= 0 → fn+1i − fniht+12α(fni+1 − 2fni + fni−1h2x+fn+1i+1 − 2fn+1i + fn+1i−1h2x)= 0 .(A.21)The FDM of this equation is illustrated in the following diagram:194A.2. Overview of Finite Difference Method✈ ✈ ✈✈ ✈ ✈× tn+1/2xixi−1 xi+1fn+1ifn+1i−1 fn+1i+1fnitntn+1and again the unknown is highlighted both in the diagram and the equation. Themain difference between this discretization and the previous one is in the fact that,this FDM requires 2 time level, whereas FDE (A.19) has 3 time levels. More im-portantly, in this scheme there are 3 unknowns in the FDA: {fn+1i−1 , fn+1i , fn+1i+1 } andtherefore there is an implicit dependency of advanced time level unknowns. Thistype of FD schemes are known as implicit schemes. The FD schemes such as theleap-frog scheme used in (A.19) – where the dependency of the FDM on the advancedtime level is explicitly a single point – are known as explicit schemes.After converting a PDE to a FDE, the next step is solving this algebraic equation.We can write an FDE in a compact form:Lhi(fn+1i , fni , · · ·) ≡ Lh (Fn+1,Fn, · · · ) = ~0 , (A.22)where Lh is the FDA operator, the most advanced time level values, fn+1i , is con-sidered as the unknown, and the superscript h denotes the typical step size of thediscretization. Here we defined the vector:Fn+1 ≡ [fn+1i ] . (A.23)Depending on the PDE and the chosen FDA scheme, this equation can be solvednumerically using various methods. For a linear PDE and an explicit scheme,195A.2. Overview of Finite Difference MethodEq. (A.22) is indeed a linear equation:AFn+1 = b (A.24)where A is a diagonal matrix and b is a vector that depends on previous time levelvalues of the function, Fn,Fn−1, · · · . In this case, solving the FDE simply reducesto inverting a diagonal matrix, i.e. inverting the diagonal terms – which can bedone in a single (trivial) matrix operation. But in general, if the PDE is linear andFD scheme is implicit, the FDE reduces to the same linear equation as (A.24), butthe matrix A is no longer diagonal. In even more general case, where the PDE isnon-linear, and the FD scheme is implicit, one needs to solve a non-linear algebraicequation for a vector of unknowns. Such systems are perhaps the most interestingand are the subject of study with the FD toolkit.In this scenario, one can solve the non-linear FDE using the multivariable iter-ative Newton method:Fn+1l+1 = Fn+1l − J−1(Rl) (A.25)in which the sub-subscript l index’s the number of Newton method iterations, i.e.Fn+1l+1 is the new approximate solution after a single iteration, and Fn+1l is the oldsolution. In recursive Eq. (A.25), J−1 is the inverse of the Jacobian matrix of the FDoperator Lh as a function of Fn+1. More explicitly, it is the multivariable derivativeof the nonlinear FDA operator L:Jji ≡ ∂Lj∂fn+1i. (A.26)Finally, in Eq. (A.25), lR denotes the “residual” of the FDE for the previous ap-proximate solution generated from the Newton iteration:Rl ≡ L(Fn+1l). (A.27)196A.2. Overview of Finite Difference MethodNote that this iterative method requires an initial guess that is usually taken to bethe previous time step solution:Fn+10 = Fn . (A.28)Here the logic is simple: if the PDE evolves the function slowly in time, Fn+1 is closeto Fn and thus Fn should be a good initial guess for it. Note that in this method,each time level update demonstrated in (A.18) has another layer of Newton iterationpresented in (A.25). This internal iteration usually converges very quickly (in fewsteps).So far, we have only provided a formal description of solving a non-linear FDE.Practically, the numerical inversion of J is a non-trivial task. One can use the Gauss-Seidel or Jacobi methods to find the inverse matrix iteratively, however, since thisJacobian is going to be used in the Newton iteration (A.25) rather than performingtwo independent iterative schemes, one can simply find an approximate inverseJacobian by only taking the diagonal part of this matrix and use that in the Newtonsolver. 43 This approach is called point-wise Newton Gauss-Seidel method and isequivalent to assuming that the only unknown in FDE is fn+1i (fixing the rest ofadvanced time level values that occur in an implicit FDA scheme) and solve for itusing a single variable Newton method:[fn+1i ]l+1 = [fn+1i ]l − [Rii]l/Jii (A.29)where:[Rii]l = Li([fn+1i ]l)(A.30)43The convergence of such method is guaranteed if the Jacobi matrix is diagonally dominant, i.e.∑i=j|Aij | >∑i6=j |Aij |197A.2. Overview of Finite Difference Methodis the residual of the FDA equation at the point i and:Jii =∂Li(fn+1i , fni , · · ·)∂fn+1i(A.31)is the diagonal element of the Jacobian matrix. Note that there are two iterationsinvolved here, one over index i, the numerical grid, and one on the Newton iterationindex l. It is ineffective to perform the l iteration first, since a highly accuratesolution to the point-wise Newton problem will become completely disrupted as soonas the value of the next neighboring point fn+1i+1 is changed via the next Newtoniteration. Therefore, it is much more effective to perform the iterating over thenumerical grid first. This is known as a single point-wise Newton Gauss-Seidelrelaxation sweep and if it converges, it usually only takes few iteration. Performingthis relaxation, for few times, a single time step evolution is complete and thealgorithm (A.18) can proceed to the next step.This algorithm is the first approach to solve a non-linear PDE and is the default(and at the moment only) method that is built into FD toolkit for solving the PDEs.As we will discuss in detail, invoking the procedure:A Gen Solve Code(DDS,{solve for var},input="d/c*",proc name="my proc")will create a low level (Fortran) routine that performs the relaxation sweep. Havingthis routine, a PDE can be solved by a driver routine that applies the relaxation asneeded (depending on some stopping criteria). Of course, solving a PDE involvesseveral other steps, such as dealing with boundary points where rather than FDAequivalent of PDE, a boundary condition needs to be imposed. This is done bydefining and passing the DDS variable which is a Maple data type to specify a PDEand its boundary conditions over a discretized domain. It is the description of theDDS and other tools and objects that are needed before applying this procedure thatconstitutes the majority of this documentation.We note that a similar discussion to what we described about the time depen-198A.2. Overview of Finite Difference Methoddent PDEs also applies to the boundary value problem PDE’s (elliptic PDEs). Forexample, consider the following second order discretization of the Laplace equation:∂2f(x, y)∂x2+∂2f(x, y)∂y2= 0 → fi+1,j − 2fi,j + fi−1,jh2x+fi,j+1 − 2fi,j + fi,j−1h2y= 0 .(A.32)The finite difference molecule for this FDA is illustrated in the following diagram:✈ ✈ ✈✈ ✈ ✈✈ ✈ ✈xixi−1 xi+1fi,jfi−1,j fi+1,jfi,j−1fi,j+1yj−1yjyj+1In this case, one needs to provide the discrete values of the function at theboundary points, and the unknowns are all of the values in the interior points fij:(BVP) {f1,j fNx,j fi,1 fi,Ny} → fi,j (unknown) (A.33)Again, a simple approach to solve this PDE is to use iterative schemes. For example,one can solve the FDA equation (A.32) for the mid-point value fij, assuming thevalues at the neighboring points are fixed. Then performing this point-wise solverprocess over all of the interior points (a relaxation sweep) iteratively will decreasethe residual to the desired tolerance (if it converges). However we note that relax-ation schemes for boundary value problems (BVP) converge very slowly and otheralgorithms such as multigrid [33] are essential to efficiently solve elliptic-type PDEs.199A.2. Overview of Finite Difference MethodA.2.3 Testing Facilities: Convergence and IREFinding a solution to a PDE or an ODE can be a complex task. However, if thesolution is given as a discrete function, checking that it satisfies the equation issomewhat a straightforward process. Consider the equation:L(f) = 0 , (A.34)where L is a differential operator and f is the unknown function. One can use anFDA scheme to discretize the differential operator:L→ Lh (A.35)where h denotes the typical “size” of the discretization. Then for a given solutionfunction, f˜ , one can evaluate the residual:Rh = Lh(f˜) (A.36)to confirm if the function f˜ satisfies the discretized version of the equation. Asolid testing facility for a numerical solver is to independently develop this residualevaluator, which we refer as Independent Residual Evaluator (IRE). Of course, theresidual (A.36) will not be exactly zero since Lh is an approximation to L andperhaps f˜ is also a numerical solution to (A.34) that differs from the exact solutionf . However, one would expect if the solution f˜ is well resolved, is “close enough”to the exact solution, and FDA operator Lh is a “good” approximation of L, thenthe norm of this residual should be orders of magnitude smaller than the actualnorm of the function f˜ . A more rigorous definition of all these concepts and howto validate the numerical solution using an IRE test will follow. However, beforethat we momentarily dive into FD toolkit and how it provides a rapid work-flow tocreating IRE routines.200A.2. Overview of Finite Difference MethodConsider the following ODE for a(x) on a given time t:da(x)dx− 1− a(x)22x− 12x[(∂φ(t, x)∂x)2+(∂φ(t, x)∂t)2]= 0 (A.37)where φ(t, x) is a time dependent field which can have its own dynamical PDE. Herewe want to evaluate the left hand side of the equation for the given discrete solutionsai and φni and verify that it is zero (numerically). The process involves creating anFDA of this ODE, evaluating the residual over the numerical domain, summing upthe point-wise residuals and returning a norm of it. FD toolkit provides an almostfully automated mechanism to do so. For example, if we use FD’s default FDAscheme (second order accurate and centered), the Maple code to generate the IREFortran routine in this case is:> read "FD.mpl": Make_FD():> grid_functions:={a,phi}:> res_a := diff(a(x),x)/a(x) - (1 - a(x)^2)/(2*x) -1/2*x*(diff(phi(t,x),x)^2+diff(phi(t,x),t)^2):> Gen_Res_Code(res_a,input="c",proc_name="ire_a");Fortran Code is written to ire_a.fC header is written to ire_a.hC call is written to ire_a_callExample 2: Creating testing (IRE) routines with FD is fully automated.The steps in this examples are: loading the FD package, initializing the internalvariables of FD, defining symbols ’a’ and ’phi’ as grid functions, writing down theODE, and passing the equation in its continuum form to the procedure:Gen Res Code(expr,input="c*/d",proc name="myproc");This call creates 3 source code files:• ire a.f: is the Fortran subroutine that evaluates the residual (A.37). Thissubroutine has the following header:201A.2. Overview of Finite Difference Methodsubroutine ire_a(a,n_phi,nm1_phi,np1_phi,x,Nx,ht,hx,res)and as you can see, it requires passing in the function a and 3 time levels offunction φ, denoted by n phi (current time), nm1 phi (retarded time), andnp1 phi (advanced time) since these values are required to compute the timederivative expression in the residual (in centered scheme). The last parameterres is a generic name, that always stores and returns the result of the com-putation (it will correspond to the updated value of the dynamical functionwhen solver routines are generated).• ire a.h is the C header (wrapper) file that needs to be included in a C driverroutine to use the subroutine, the content of this file is:void ire_a_(double *a,double *n_phi,double *nm1_phi, double *np1_phi,double *x,int *Nx,double *ht,double *hx,double *res);• ire a call: is a plain text file containing a typical C call of the routine. callfiles can be copied to a C driver code. For example, here the content of thefile is:ire_a_(a,n_phi,nm1_phi,np1_phi,x,&Nx,&ht,&hx,res);which as you can see, is a C call with the last parameter, again, labeled asres. After copying the content of call to the driver code, the user needs toappropriately change the name of the last parameter to the allocated vector(pointer) or the single variable defined in the C driver to store the result. 44In this example, the result, res, is a number (a double precision floating pointnumber) containing the norm of the residual. FD also assumes that in the Cdriver, the user will define the name of the allocated vectors and parametersfor the PDE similar to what they are defined in the Maple expression.44Of course, a good strategy is to avoid naming any variables in the C driver code as res. Thename res does not need any modification in the Fortran routine or C header file.202A.2. Overview of Finite Difference MethodWe will discuss this procedure and similar other code generator procedures in moredetails through Sec. A.3 to Sec. A.6. Following note is a mathematical discussionon the notion of convergence and independent residual evaluators. Even though,these concepts are crucial to validate the consistency and accuracy of the numericalsolver, the following is somewhat independent of the FD toolkit and applies to anyfinite difference method. This manual should be accessible without expertise in themathematical discussion in the following note.· · ·Note on convergence and IRE testsConsider that the solution in Eq.(A.34) is produced by solving a finite differenceapproximation for the PDE. To preface this section, we first review our notation:L(f) = 0 (A.38)Sh(fh) = 0 (A.39)Lh(fh) = Rh (A.40)i.e. L is the PDE operator in continuum form, and f is the continuum solution,fh is the numerical solution and Sh is the solver FDA (the FDA of the originalPDE that is used in the numerical solver). Finally, Lh is another FDA to L thatis different than Sh, and due to this difference the RHS is nonzero and symbolizedby the residual Rh. Note that previously we used Lh to denote the FDA used inthe numerical solver, but here we are mostly interested in testing the solver using adifferent FDA operator which is the main focus of this section and thus denoted byLh.If the numerical solution fh is convergent at the continuum limit – where the203A.2. Overview of Finite Difference Methoddiscretization size h approaches zero– we denote the continuum limit by u:∃u = limh→0fh (A.41)therefore one can assume the following Richardson expansion:fh = u+ ehf = u+ e1h+ e2h2 + · · · (A.42)where the coefficients e1, e2 are functions independent of h. As one might expect,the error in the solution ehf depends on the accuracy of FDA Sh that is used in thenumerical solver. The first non-zero coefficient ep that appears in the expansiondefines the accuracy of the solution, and is the dominant part of the error in thelimit h→ 0. For example, a second order convergent solution has the form:fh = u+ e2h2 + · · · (A.43)and using this expansion it is easy to show that for the 3 consecutively refinedconvergent solutions: fh, fh/2 and fh/4 the limit of the following ratio:limh→0Q =||fh − fh/2||||fh/2 − fh/4|| = 4 , (A.44)is 4. Here ||.|| is some norm of a discretized functions. Measuring the factor Q isreferred as standard convergence test in the literature.For a convergent numerical solution fh, it is not clear that the limiting continuumfunction u (A.41) is indeed the solution to the continuum problem L(f) = 0, i.e. wewant to know if:u?= f . (A.45)To further emphasize this: the numerical solution fh might be convergent but weneed some sort of proof to show that it is in fact converging to the correct solution.204A.2. Overview of Finite Difference MethodOne might speculate that this should be the case if1) Sh approximates L correctly, or more rigorously:limh→0Sh = L (A.46)known as consistency condition condition for the finite difference scheme.2) The method used to solve the finite difference equation is stable. We refer thereader to [95] for mathematical definition and discussion on the notion of stability.In certain cases (for linear PDEs) it can be proven that stability and consistency aresufficient conditions for convergence. However, to our knowledge, there is no suchproof for non-linear cases which most of the interesting physical systems exhibit.We also note that from a practical point of view there is no simple prescription orcondition that can be checked off to ensure the stability of the method for non-linearsystems.Here we rather take a practical approach: the independent residual evaluationtest. The IRE test provides a stronger test than the standard convergence test,and validates (or rejects) the equality A.45. Suppose that fh is O(hp) convergent,meaning:fh = u+ ehfehf = ephp + o(hp) (A.47)where ep is a function, independent of h and o(hp) is an h dependent function thatconverges to zero faster than hp:limh→0||o(hp)||hp= 0 (A.48)Now suppose, as defined in the beginning of this discussion in Eq. A.40, Lh isanother FDA of the original continuum operator L (created with a different FD205A.2. Overview of Finite Difference Methodscheme than Sh and is also created independently). Lh is what we refer as inde-pendent residual evaluator. We assume that this operator is consistent with thecontinuum operator L upto accuracy O(hq), meaning:Lh(g) = L(g) + ehL(g)ehL(g) = hqEL(g) + oL(g;hq) (A.49)where EL is an h independent operator, and oL(.;hq) is an h dependent operatorwith a norm that converges to zero faster than hq:limh→0||oL(g, hq)||hq= 0 (A.50)Note that here we are assuming that the operator expansion (A.49) is possible forthe function g. Intuitively, one would expect this assumption to hold for functionsthat are well resolved over the discretized domain. Particularly in the case of g = fh,this is a plausible assumption, as we expect the numerical solver to produce a well-resolved discrete solution.Now the claim is that if the conditions (A.47) and (A.49) hold then the residualdefined as:Rh ≡ Lh(fh) (A.51)converges to zero if and only if fh is indeed converging to f , the continuum solution,i.e.:u = f (A.52)Furthermore the convergence behaviour of the residual is dominated by the twoerrors: the solution fh error, which we assumed to be O(hp) and the error of the Lhoperator which we assumed to be O(hq) and is explicitly of the form:||Rh|| = O(hp) +O(hq) = O(hmin(p,q)) (A.53)206A.2. Overview of Finite Difference MethodTherefore, for example if both the solution and the IRE are second order convergentthen, one would expect to observe a second order convergence in the residual Rh aswell.Linear case:We first prove the claim for the linear operators L and Lh:Lh(fh) = Lh(u+ ehf ) = Lh(u) + Lh(ehf ) = L(u) + ehL(u) + L(ehf ) + ehL(ehf )= L(u) + hqEL(u) + hpL(ep) + hqhpEL(eq) + · · · = L(u) +O(hq) +O(hp) + · · ·(A.54)where · · · are higher order terms. We used the fact that L, Lh and ehL are linearoperators (the linearity of ehL follows from its definition (A.49)). Note that in theexpansion of the term Lh(ehf ), we are assuming that the error function ehf is alsowell resolved function on the mesh such that the expansion (A.49) is meaningful.Nonlinear case:In the non-liner case, a similar analysis can be performed by linearizing the FDAoperator Lh. We assume that Lh is differentiable around g, meaning there exist alinear operator DhL[g] such that:Lh(g + q) = Lh(g) +DhL[g](q) + ohL[g](q) (A.55)and ohL[g] is an operator with a norm converging to zero faster than ||q||:lim||q||→0||ohL[g](q)||||q|| = 0 (A.56)The differential operator DhL[g] can be naively defined as the limit:DhL[g](q) ≡ limǫ→0Lh(g + ǫq)− Lh(g)ǫ(A.57)Note that the differentiability of Lh is simply guaranteed if all of the partial deriva-tives ∂Li(g)/∂gi˜ exist where Li is the FDA equation at the point indexed by i and gi˜207A.2. Overview of Finite Difference Methodis the discrete value of the function at the point indexed by i˜. 45 These derivativesobviously exist for normal FDA operators used in finite difference methods. Wealso note that the abstract DhL[g] operator in a matrix representation is simply the∂Li(g)/∂gj matrix. Furthermore, not surprisingly, it is equal to the FDA operatorLh itself, when Lh is linear:DhL[g](q) =Lh(g + ǫ q)− Lh(g)ǫ=ǫLh(q)ǫ= Lh(q)⇒ DhL[g] = DhL = Lh (A.58)Note that in linear case, DhL indeed does not depend on g anymore, as the operatorLh. Assuming the differentiability of Lh around u, we have:Lh(fh) = Lh(u+ ehf ) = Lh(u) +DhL[u](ehf ) + ohL[u](ehf )= L(u) + ehL(u) +DhL[u](ehf ) + ohL[u](ehf )= L(u) + hqEL(u) + oL(u;hq) +DhL[u](ephp + o(hp)) + ohL[u](ephp + o(hp))= L(u) + hqEL(u) + hpDhL[u](ep) + o(hp) + o(hp) (A.59)where in the last step we used the linearity of DhL[u] and the property of ohL[u]operator (A.56). This result again translates to:Lh(fh) = L(u) +O(hp) +O(hq) = L(u) +O(hmin(p,q)) (A.60)and the residual Lh(fh) will converge to zero, if and only if L(u) = 0, or u thecontinuum function that the numerical solution fh is converging to, is indeed theunderlying continuum solution f .Now using this result we have a stronger test: The convergence of the IRELh(fh) is only possible if the solution is convergent and is converging to the correct45Note that here i and i˜ can be any of the discrete domain indices, here we are simply using i asa symbol of discretization208A.3. Semantics of FDsolution. Therefore if one can create a solid IRE operator Lh that is consistent withL, checking the convergence of the IRE will guarantee the accuracy of the solution.Of course, one can ask: what if Lh also has an error in its implementation ? Herethe keyword independent development becomes crucial. If the independent residualis converging, it is extremely unlikely that Sh and Lh that should be developedcompletely independently will both have an internal error, and both of the errorsagree, i.e both Sh and Lh happen to be identical to an FDA for another PDE thatis not the original PDE. Often it is best to create the IRE operator Lh using anautomated process to reduce possible human errors. This, in part, was the originalmotivation to develop FD and as it will be discussed further, generating IRE routinesis been fully automated in FD toolkit.A.3 Semantics of FDIn this section, we describe some of the internal variables of FD and two derivedalgebraic data types that FD uses to work with finite difference expressions.A.3.1 Parsing a PDE: Fundamental Data TypeAs mentioned in the introduction, FD is developed with the philosophy that user’sinvolvement in the straightforward tasks should remain minimal. Consider the fol-lowing PDE for f :∂tf(t, x, z) + β(t, x, z)∂xf(t, x, z)+γ(x)∂zf(t, x, z) + a∂2xf(t, x, z) + b∂2zf(t, x, z) + g(x, z) = 0 (A.61)The LHS written in canonical Maple form (without use of aliases) is:PDE:=diff(f(t,x,z),t) + beta(t,x,z)*diff(f(t,x,z),x)+gamma(x)*diff(f(t,x,z),z) + a*diff(f(t,x,z),x,x)+b*diff(f(t,x,z),z,z)+ g(x,z);209A.3. Semantics of FDOne can easily observe that this expression, by itself, contains enough informationregarding the dimensionality of the problem, functions and their dependencies, pa-rameters, and of course derivatives. By looking at the expression, we can concludethat:• f is a time dependent function, defined on a 2 dimensional spatial domainlabeled by (x, z).• β is also time dependent with same spatial domain as f .• g is a time independent function only defined on the (x, z) domain.• γ has only 1 dimensional dependency on x coordinate.• a and b are parameters (assuming that all dependencies are explicitly pre-sented)• the order and direction of derivatives of f are clear.There is no need for further specification to implement this PDE on a computer. Thefirst step to reducing potential human errors is to eliminate another intermediatesyntactic language to write a PDE. Rather, FD uses Maple’s powerful symbolicmanipulation capabilities and has a built-in parser which allows directly passing aPDE to its routines. This puts the entire complexity of the fundamental data typeon the expression, and frees the user from providing any further specification. Assoon as an error-proof PDE is written down, (which is easily possible as the workingenvironment of FD is Maple with all its symbolic tools) the task of identifyingthe parameters, functions, dimensionalities, derivatives, and required time levels toperform FDA in time dimension is left to the software. This is one of the advantagesof FD, over previously developed software such as RNPL [114]. This also makes FDan efficient prototyping language, particularly for developing testing facilities as wedemonstrated in Example 2.210A.3. Semantics of FDA.3.2 CoordinatesFD reserves the variables (t,x,y,z) for the name of the time and spatial coor-dinates that define the domain of a PDE. They are protected variables after FDis loaded. Similarly, FD reserves the symbols (n,i,j,k) for indexing the corre-sponding coordinate points (t(n),x(i),y(j),z(k)). It uses (ht,hx,hy,hz) asthe name for the step-size of the discretization along these coordinates, respectively.The names (Nt,Nx,Ny,Nz) are reserved for the size of the discretized domain, and(xmin,xmax), (ymin,ymax), (zmin,zmax) are reserved for flags to specify the in-ner CPU boundary points of the coordinates (their applicable is in the context ofparallelization).This association can be demonstrated as:t↔ n↔ ht ↔ Ntx↔ i↔ hx ↔ Nx ↔ (xmin, xmax)y ↔ j ↔ hy ↔ Ny ↔ (ymin, ymax)z ↔ k ↔ hz ↔ Nz ↔ (zmin, zmax) (A.62)and is built into FD. The coordinate names, and this association table are neces-sary to identify functions, differential expressions, and perform finite differencing.For example, FD recognizes that an expression such as f(x+hx,y-2*hy) should bediscretized as f(i+1,j-2), or an expression such as f(x+hy) is invalid and can-not be discretized, since hy is not an stepping size in x direction. Ultimately, thisassociation table allows FD to discretize a differential expression such as ∂xf(x, y)(in Maple notation: diff(f(x,y),x)), directly to (f(i+1,j)-f(i-1,j))/(2*hx)without any need for further specification. (See the example in Sec. A.3.4).211A.3. Semantics of FDA.3.3 Initializing FD, Make FD, Clean FDAs the reader may have noticed from the previous examples, FD is in a Maple scriptformat, and can be imported to a Maple worksheet/script by executing:read("/your/fd/directory/FD.mpl");FD’s internal variables are initialized by calling the procedure:Make FD();which has a short alias, MFD(), and creates the table for the coordinate associationdescribed in Eq. A.62) and initializes the default finite difference table that specifiesthe finite difference scheme. We will further discuss this table in Sec A.4.2. To cleanthe initialized variables, user can execute:Clean FD();or use the alias CFD().A.3.4 Grid Functions Set: grid functionsFD uses a global variable named grid functions (of type set in Maple) as itsreference for all of the function names that are expected to be discretized as:f(t, x, y, z)→ f(tn, xi, yj, zk) ≡ fni,j,k . (A.63)In Maple language, if symbol f is in the grid functions, then the function f(t,x,y,z)(in its most generic 1+3 dimensional case) will be converted to f(n,i,j,k) duringthe process of discretization. The following example demonstrates how FD uses thecoordinate names, the coordinate association table, and the symbols defined in gridfunctions to produce FDA expressions:212A.3. Semantics of FD> read "FD.mpl": MFD():> grid_functions:={f}:> Gen_Sten(f(t,x,y,z));f(n, i, j, k)> Gen_Sten(diff(f(x,y),x));f(i - 1, j) - f(i + 1, j)-1/2 -------------------------hx> Gen_Sten(x+g(y,z));x(i) + g(y(j), z(k))Example 3: Discretization of grid functions vs non-grid functionsHere, Gen Sten is the main routine that performs the finite differencing and willbe discussed extensively. However, user can easily guess its functionality from theexample. As it can be seen, beside the names that are included in the grid functionset, the coordinate variables (t,x,y,z) are by definition grid functions and arediscretized as (t(n),x(i),y(j),z(k)). Furthermore, if a symbol with coordinatedependency (such as g(y,z) above) is not included in the grid functions set, it willbe considered as an external function that user will provide to the Fortran routines.FD discretizes its coordinate functions rather than the function. For example, hereit is discretized as: g(y(j),z(k)) rather than g(j,k).Time Level Reduction:We shall emphasize that the discrete expression f(n,i,j,k) will be eventually (atthe point of code generation) replaced by: n f(i,j,k). This process is done in-ternally, and is referred as time level reduction (See Sec. ??). The time level n isusually referred as current time level, n-1 is referred as retarded time level and n+1 iscalled advanced time level. When FD performs the time level reduction, it uses theprefix np1 and nm1 in the names of the advance and retarded time levels functionsrespectively:f(n,i,j,k)→ n f(i,j,k)f(n+1,i,j,k) → np1 f(i,j,k)213A.3. Semantics of FDf(n-1,i,j,k) → nm1 f(i,j,k)The higher time level f(n+2,i,...) will be renamed to np2 f(i,...) and thesyntax for the other cases should be clear. This replacement is simply because intime dependent finite difference algorithms only a finite number of time levels areneeded and stored in the memory during the time evolution. The user can definethe time levels in the C driver code according to this standard, or can define “alias”pointers (that adopts these names) to the underlying data structure to be able touse the FD generated routines.A.3.5 Known FunctionsFD has a set of “known” functions, which is basically a set of floating point functionsthat are known to the low level language (Fortran here). These functions in FD are:{ln,log,exp,sin,cos,tan,cot,tanh,coth,sinh,cosh,exp,sqrt,‘∧‘,‘*‘,‘+‘,‘-‘,‘/‘}and during a discretization process, FD does not convert their arguments to adiscrete version, rather it discretizes the arguments accordingly. For example,sin(z)*f(x)+exp(y) will be discretized as:> Gen_Sten(sin(z)*f(x)+exp(y));sin(z(k)) f(i) + exp(y(j))assuming that f is in grid functions.A.3.6 Valid Continuous Expression, VCEValid Continuous Expression (VCE) is an algebraic function of the continuous co-ordinate variables, (t, x, y, z), in which the dependencies of grid functions on thecoordinates are only of the form:f(t+ lht, x+mhx, y + qhy, z + phz) , (A.64)214A.3. Semantics of FDwhere (l,m, q, p) are known integers (not variable), and (ht, hx, hy, hz) are the as-sociated stepsize variables. Furthermore, a VCE does not have explicit dependencyon the discretization indices (n, i, j, k). For example, if functions f and g are gridfunctions, then all of the expressions:f(t,x,y) + (g(x+hx)-g(x-hx))/(2*hx)r(x*y)u(sin(x*y),g(z))f(x+2*hx,y-3*hy)/hz + x*z^2 + g(z,x,t,y)are VCE, andf(t,x+hy)g(x,y+2)f(x(i),y(j))cos(j)f(u(x),y)g(x*y)diff(f(x),x)are all invalid continuous expressions. Particularly, compare g(x*y) and r(x*y),former is not VCE, since g is defined as a grid function, while later is a VCE as r isconsidered an external function. Note that FD does not check for the consistencyin the order of the variables, i. e. f(x,y) + f(y,x) is considered a VCE.A.3.7 Valid Discrete Expression, VDEValid Discrete Expression (VDE) is an expression in which the explicit dependenciesof functions on the discretization indices (n,i,j,k) is only via the grid functions orcoordinates. Furthermore, this dependency is of the form: f(n+ q, i+m, j+p, . . . ),where q,m, p, · · · are known integers, and f is either a grid function or is one of the215A.3. Semantics of FDcoordinates (t, x, y, z). In the case of coordinate, indexing must be done according tothe coordinate-index association (A.62). For example, for grid functions:={f,g}:g(i+1,j-2)x(i)u(x(i),f(j,k),a)f(j,k+2,i)are all VDE and,y(i)x(i)+ksin(i)f(i*j)u(i)f(i,y(j))are invalid discrete expression.A.3.8 Conversion Between VDE and VCEThe definition of VDE and VCE allows a one-to-one mapping between these twotypes. FD provides two functions for the conversion:A:=DtoC(B::VDE);B:=CtoD(A::VCE);Even though VCE’s are not practically useful for numerical implementations, theconversion of a VDE to VCE can be used for demonstration and testing purposes.For example, a finite difference expression in VDE form, can be converted to a VCE,and then a Taylor expansion of it can reveal its equivalent continuum differentialoperator. The following demonstrates the process for Kreiss-Oliger dissipation op-erator [95] that is commonly used in finite difference methods:216A.4. Discretizing a PDE> read "FD.mpl": Make_FD():> grid_functions:={f}:> A:= -epsdis/(16*ht)*( 6*f(n,i) + f(n,i+2) + f(n,i-2)-4*(f(n,i+1) + f(n,i-1)) ):> B:=DtoC(A):> E:=convert(series(B,hx),polynom);4epsdis D[2, 2, 2, 2](f)(t, x) hxE := -1/16 ---------------------------------htExample 4: Conversion between VDE and VCEwhich gives:E =−ǫ16(∂4f(t, x)∂x4)h4xht(A.65)A.4 Discretizing a PDEIn this section we discuss how to perform a finite differencing on a PDE using thefacilities of FD, how to choose a specific discretization scheme and and how to accessthe results of a lengthy finite difference operation.A.4.1 Performing the Finite Differencing, Gen StenThe main routine that performs FDA is:VDE/VCE::Gen Sten(expr)(with an alias: GS) where the expr is an arbitrary mixed differential/algebraic Mapleexpression. As mentioned before, this routine performs the discretization on the gridfunctions and coordinates, leaving parameters and other functions unchanged ( thecoordinate of the functions however will be discretized). The result is by default aVDE type. To return a the finite difference expression in VCE form, the optionalinput discretized should be disabled:217A.4. Discretizing a PDEVCE::Gen Sten(expr,discretized=false)Note: In the examples in the rest of this manual, we assume that the FD initial-ization is invoked and f and g are grid functions:> read "FD.mpl": MFD():Warning, grid_functions is not assignedFD table updated, see the content using SFDT() command> grid_functions:={f,g}:Here is an example of discretizing differential expressions:> A:=diff(f(x,y),x,y):> B:=Gen_Sten(A);-f(i - 1, j - 1) + f(i - 1, j + 1) + f(i + 1, j - 1) - f(i + 1, j + 1)B := -1/4 ----------------------------------------------------------------------hy hx> Gen_Sten(diff(f(x),x)+g(y)+cos(f(x))+r(x)+z);f(i - 1) - f(i + 1)-1/2 ------------------- + g(j) + cos(f(i)) + r(x(i)) + z(k)hx> Gen_Sten(A,discretized=false);-f(x - hx, y - hy) + f(x - hx, y + hy) + f(x + hx, y - hy) - f(x + hx, y + hy)-1/4 ------------------------------------------------------------------------------hy hxExample 5: Discretizing a PDEAs one can see, the default discretization scheme in FD is centered (and secondorder accurate). In the next section, we describe how to change the finite differencescheme.A.4.2 Discretization Scheme, FD tableFD uses an internal table, FD table, to perform the finite difference operations suchas ones in Example 5. This table, simply is a list of the points that can be used218A.4. Discretizing a PDEfor the n’th derivative computation for each of the coordinates (t,x,y,z). Forexample, the x component of this table is:> FD_table[x];[[0], [-1, 0, 1], [-1, 0, 1], [-2, -1, 0, 1, 2], [-2, -1, 0, 1, 2], ...where n’th element (counting from zero), is a list of points specifying the finitedifferencing scheme for the n’th derivative along x. The numbers present the list ofneighboring points to x(i) that are allowed to be used for FDA. For instance, thethird element, [-2,-1,0,1,2], presents the 5 points: central point x(i), 2 to leftand 2 to right, that are allowed for FDA of the third derivatives in x coordinate.This is demonstrated in the following diagram.×❡ ❡ ❡ ✉ ❡ ❡ ❡×0↓x(i-2) x(i-1) x(i) x(i+1) x(i+2)→ → . . .←←. . .1 2−1−2[-2,-1,0,1,2]Figure A.1: Five points specifying the FDA scheme for the third derivatives inFD table in x direction.FD initializes the finite difference table when Make FD() is invoked. By default,the table is upto the 5’th derivatives (adjustable by the global variable MAX DERIVATIVE NUMBER=5)in all dimensions, and the points expand symmetrically around the central point(centered finite differencing).A.4.3 Changing the FDA Scheme: FDS, Update FD TableThe FD scheme can be chosen by adjusting the content of FD table. FD providesa convenient routine for this purpose:Update FD Table(order::integer,fds::FDS);219A.4. Discretizing a PDEin which the user specifies the desired order of accuracy, order, and the schemevia the second argument fds. This argument is a table with a particular formatwhich we refer as a Finite Difference Specifier (FDS). A FDS is a table for the 4coordinates, (t,x,y,z),fds:=table( [ t=... , x=..., y=... , z=...] );and each element has the following format:X = [p_left,-1] or [-1,-1] or [-1,p_right]in which X denotes one of the coordinates, and the values p left and p right areknown integers. p left specifies how many points to left of the central point isallowed, and similarly p right specifies the number of points to the right that canbe used in an FDA for coordinate X. If these values are set to -1 it allows FD toexpand in that direction to as many point as needed to achieve the desired accuracy.At least one of the p-values must be set to -1. Particularly, the p left and p rightneed to be adjusted for creating FDAs that can be applied in the vicinity of theboundaries of the numerical grid. This is demonstrated in the following diagram:✉ ❡ ❡ ❡ ❡ . . . fds:=table([ t=[-1,-1],x=[0,-1],y=[-1,-1],z=[-1,-1] ]):❡ ✉ ❡ ❡ ❡ . . . fds:=table([ t=[-1,-1],x=[1,-1],y=[-1,-1],z=[-1,-1] ]):❡ ❡ ✉ ❡ ❡ . . . fds:=table([ t=[-1,-1],x=[2,-1],y=[-1,-1],z=[-1,-1] ]):. . . ❡ ❡ ✉ ❡ ❡ . . . fds:=table([ t=[-1,-1],x=[-1,-1],y=[-1,-1],z=[-1,-1] ]):. . . ❡ ❡ ✉ ❡ ❡ fds:=table([ t=[-1,-1],x=[-1,2],y=[-1,-1],z=[-1,-1] ]):. . . ❡ ❡ ❡ ✉ ❡ fds:=table([ t=[-1,-1],x=[-1,1],y=[-1,-1],z=[-1,-1] ]):. . . ❡ ❡ ❡ ❡ ✉ fds:=table([ t=[-1,-1],x=[-1,0],y=[-1,-1],z=[-1,-1] ]):Figure A.2: Specifying different types of FD schemes: note the values in the high-lighted color and how it associates with each case at the vicinity of the boundary inx direction.220A.4. Discretizing a PDEWe remind the reader that higher derivatives require more points. In addition,increasing the accuracy order also adds to the number of the points used in FDAs.The routine Update FD Table has a built-in function P (n,m)∂m∂Xmwith O(hn) accuracy→ P (n,m) (A.66)for each of the forward, backward and centered schemes that estimates the minimumnumber of points required to achieve the desired accuracy (or better).For example the following code updates the FD table use FD scheme forward intime, centered in x, backward in y and asymmetric backward in z, with 4’th orderaccuracy. The resulting FD table is demonstrated by inspecting each element of it:> fds:=table([t=[0,-1],x=[-1,-1],y=[-1,0],z=[-1,2]]):> Update_FD_Table(4,fds):FD table updated, see the content using SFDT() command> FD_table[t];[[0], [0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5, 6],...]> FD_table[y];[[0], [-4, -3, -2, -1, 0], [-6, -5, -4, -3, -2, -1, 0], ...]> FD_table[z];[[0], [-2, -1, 0, 1, 2], [-4, -3, -2, -1, 0, 1, 2], [-4, -3, -2, -1, 0, 1, 2],...]Example 6: Changing the Finite Difference SchemeWe note that FD table can be updated manually by overwriting the elements, how-ever this method is error-prone, and higher derivatives in particular might not havea sufficient number of points to be evaluated as a FDA. For example, in the follow-ing, we specify only 2 points for the second derivative in time, and the Gen Stenprocedure outputs an error as it is impossible to compute the FDA equivalent of theinput according to the FD table:221A.4. Discretizing a PDE> FD_table[t]:=[[0],[0,1,2],[0,1]]:> Gen_Sten(diff(f(t,x),t));-f(n, i) + f(n + 1, i)----------------------ht> Gen_Sten(diff(f(t,x),t,t));Error, (in Calc_Stencil_L) Failed to find FDA coefficients, check FD_table content!Finally, we note that the entire content of FD table (rather lengthy sequence ofintegers!) can be viewed using the procedure:Show FD Table();A.4.4 Accessing the FD Results: Show FDIf the Gen Sten procedure is used to perform finite differencing on a lengthy differ-ential expression, the resulting FDA is not human readable. To better present whatGen Sten has performed, the routine stores the differential expressions it finds inthe input and their FDA equivalent that it replaces them with, in an internal tablenamed FD results. The content of this table can be accessed using the procedure:Show FD();For example, consider the following finite differencing operation:> A:=diff(y*f(x,y)*diff(sin(x*y)*g(x),x),x,y):> B:=Gen_Sten(A):memory used=11.4MB, alloc=5.4MB, time=0.59> lprint(B);1/2*(-f(i-1,j)+f(i+1,j))/hx*(cos(x(i)*y(j))*y(j)*g(i)-1/2*sin(x(i)*y(j))*(g(i-1)-g(i+1))/hx)+1/4*y(j)*(f(i-1,j-1)-f(i-1,j+1)-f(i+1,j-1)+f(i+1,j+1))/hy/hx*(cos(x(i)*y(j))*y(j)*g(i)-1/2*sin(x(i)*y(j))*(g(i-1)-g(i+1))/hx)+1/2*y(j)*(-f(i-1,j)+f(i+1,j))/hx*(-sin(x(i)*y(j))*x(i)*y(j)*g(i)+cos(x(i)*y(j))*g(i)-1/2*cos(x(i)*y(j))*x(i)*(g(i-1)-g(i+1))/hx)+f(i,j)*(-sin(x(i)*y(j))*y(j)^2*g(i)-cos(x(i)*y(j))*y(j)*(g(i-1)-g(i+1))/hx+sin(x(i)*y(j))*(g(i-1)-2*g(i)+g(i+1))/hx^2)+1/2*y(j)*(-f(i,j-1)+f(i,j+1))/hy*(-sin(x(i)*y(j))*y(j)^2*g(i)-cos(x(i)*y(j))*y(j)*(g(i-1)-g(i+1))/hx+sin(x(i)*y(j))*(g(i-1)-2*g(i)+g(i+1))/hx^2)+y(j)222A.4. Discretizing a PDE*f(i,j)*(-cos(x(i)*y(j))*x(i)*y(j)^2*g(i)-2*sin(x(i)*y(j))*y(j)*g(i)-sin(x(i)*y(j))*x(i)*y(j)*(-g(i-1)+g(i+1))/hx+cos(x(i)*y(j))*(-g(i-1)+g(i+1))/hx+cos(x(i)*y(j))*x(i)*(g(i-1)-2*g(i)+g(i+1))/hx^2)# Checking if B is indeed an FDA for A:> E:=DtoC(B):> E:=convert(series(E,hx,4),polynom):> E:=convert(series(E,hy,4),polynom):> residual:=simplify(eval(A-E,hx=0,hy=0));residual := 0Expression A has several derivatives of the functions f and g that are replaced withFDA expressions. Now by invoking Show FD() we can see what replacements havebeen done:> Show_FD();d -f(i - 1, j) + f(i + 1, j){-- f(x, y) = [1/2 --------------------------, [[x, 2], [y, -1]]],dx hxd f(i, j - 1) - f(i, j + 1)-- f(x, y) = [-1/2 -------------------------, [[y, 2], [x, -1]]],dy hyd -g(i - 1) + g(i + 1)-- g(x) = [1/2 --------------------, [[x, 2]]],dx hx2d g(i - 1) - 2 g(i) + g(i + 1)--- g(x) = [----------------------------, [[x, 2]]],2 2dx hxd g(j - 1) - g(j + 1) d-- g(y) = [-1/2 -------------------, [[y, 2]]], ----- f(x, y) = [dy hy dy dx223A.4. Discretizing a PDE-f(i - 1, j - 1) + f(i - 1, j + 1) + f(i + 1, j - 1) - f(i + 1, j + 1)-1/4 ----------------------------------------------------------------------,hy hx[[x, 2], [y, 2]]] }Here the numbers next to the coordinate variables x,y denotes the order of accuracyof the replacement, and as expected they are all second order accurate. -1 representsexact FDA, i.e. there is no differentiation with respect to that coordinate. Notethat this accuracy is not what user specifies when updating FD scheme (in previoussection). It is indeed the computed value of the actual accuracy of FDA whichshould be equal or higher to the user specified value.A.4.5 Defining Manual Finite Difference Operators: FDFD provides a way to define an arbitrary FDA operator. In principal, any finitedifference operator can be created from the shifting operator (See [95]) defined (in1 dimension) as:E(fi) = fi+1 (A.67)and its inverse is simply: E−1(fi) = fi−1. The generalization of this operator isdefined in the FD toolkit, and is named FD with the following format:VDE::FD(dexpr::VDE, [ [t shift] ,[x shift,y shift,z shift] ])in which FD takes an input dexpr of type VDE, and returns a VDE that is shiftedby the given integers (t shift, x shift,y shift,z shift). If there is no timeindex dependency in the expression, the first argument, [t shift], can be droppedand the routine accepts a shorter format:FD(VDE,[x shift,y shift,z shift])Similarly if z index k does not occur in the VDE, the routine accepts shorter list[x shift,y shift] and so on. For example, the following demonstrates the defi-nition of 3 manual FDA operators: 1) a forward time derivative FDA (DT) that is224A.5. Posing a PDE & Boundary Conditions Over a Discrete Domainequivalent to ∂t upto first order accuracy, 2) centered in x derivative FDA (DXC),which is equivalent to ∂x upto second order accuracy, and 3) the time averagingoperator AVGT that is not an FDA. This operator is usually used in Crank-Nicolsonmethod to create a implicit FD scheme.> DT := f -> ( FD(f,[[1],[0]]) - FD(f,[[0],[0]]) )/ht:> df:= DT(f(n,i));f(n + 1, i) - f(n, i)df := ---------------------ht> DXC:= f -> ( FD(f,[1,0]) - FD(f,[-1,0]) ) /(2*hx):> DXC(f(i)*x(i)^2*g(j)+y(j));2 2f(i + 1) x(i + 1) g(j) - f(i - 1) x(i - 1) g(j)1/2 -------------------------------------------------hx> AVGT := f -> ( FD(f,[[1],[0]]) + FD(f,[[0],[0]]) )/2:> AVGT(Gen_Sten(diff(f(t,x),x)));f(n + 1, i - 1) - f(n + 1, i + 1) f(n, i - 1) - f(n, i + 1)-1/4 --------------------------------- - 1/4 -------------------------hx hxExample 7: Defining manual difference operatorsA.5 Posing a PDE & Boundary Conditions Over aDiscrete DomainIn solving PDEs, it often occurs that some part of the discretized domain needsspecial treatment. By its nature, boundary points require different equations thanthe original PDE. In addition, if the discretization scheme results in large finitedifference molecules, the points next to the boundaries also require special handling.For example consider 4’th order accurate FDA of the derivative of a function, ∂xf(x):225A.5. Posing a PDE & Boundary Conditions Over a Discrete Domainf(i - 2) - 8 f(i - 1) + 8 f(i + 1) - f(i + 2)1/12 ---------------------------------------------hxThis expression cannot be evaluated where i < 3 or i > Nx−2, as the finite differencemolecule (-2,-1,0,1,2) require points that do not exist in the discretized domainat these limits. In this section, we describe the methodology to create differentequations for each part of the numerical domain, and the facilities FD provides toimpose boundary conditions and implement techniques such as ghost cells.A.5.1 Discrete Domain Specifier: DDSTo specify each portion of the discrete domain, {i ∈ (1, Nx)} × {j ∈ (1, Ny)} · · · ,FD uses a syntax similar to RNPL [114], via a derived data type that we refer asDiscrete Domain Specifier (DDS). A DDS is a list of equations:DDS = [ equation1, equation2, ... ]where each equation specifies part of the discrete domain and has the following LHSand RHS:each equation: { indexeq1, indexeq2, ... } = expressionin which each expression can be a VDE, or a continuous PDE, and each indexeqdescribes the indexing for one of the spatial dimensions:each indexeq: I = [start,NI-stop,step]Here, the variable I denotes one of the indexing labels, (i,j,k), NI is the associateddomain size Nx,Ny,Nz, and step is a known integer that determines the steppingsize. The indexeq symbolizes a portion of the domain in which index I takesthe values: (start,start+step,start+2*step,...) and ends at value smaller orequal to NI-stop. The reader may notice that this is exactly equivalent to a forloop structure. For example226A.5. Posing a PDE & Boundary Conditions Over a Discrete Domain{ i = [1,Nx,1] , j =[2,Ny-1,2] } = ...is equivalent to (in Fortran syntax):DO i=1,Nx,1DO j=2,Ny-1,2...ENDDOENDDOThe following example clarifies this syntax, and demonstrates a DDS for heat equa-tion where the boundary points are fixed to values T0 and T1 and interior points arespecified by the heat equation.HeatEq:= diff(f(t,x),t) - diff(f(t,x),x,x);HeatDDS := [{ i=[1,1,1] } = f(n+1,i) - T0 + myzero*x(i) ,{ i=[2,Nx-1,1] } = Gen_Sten(HeatEq) ,{ i=[Nx,Nx,1] } = f(n+1,i) - T1 +myzero*x(i)];Example 8: 1-D discrete domain specifier for the heat equationThe necessity of myzero*x(i) expression will become clear later when we use thisDDS as an input to FD’s solver routine generator.Note that heat equation and its boundary conditions are simple and compactenough to be discretized inside the DDS. For a more complex case, it is betterto create the discrete version of the equations for the boundaries separately, andpass them into the DDS using human readable names. For example, the followingdemonstrates a 2 dimensional DDS where each boundary uses a specific discreteequation priorly created by the user:227A.5. Posing a PDE & Boundary Conditions Over a Discrete Domainmydds2d := [# Interior points:{ i=[2,Nx-1,1] , j = [2,Ny-1,1] } = EQD_interior ,# Boundaries:{ i=[1,1,1] , j = [1,Ny,1] } = EQD_left ,{ i=[Nx,Nx,1] , j=[1,Ny,1] } = EQD_right ,{ i=[1,Nx,1] , j=[1,1,1] } = EQD_bottom ,{ i=[1,Nx,1] , j=[Ny,Ny,1] } = EQD_top];Example 9: Two dimensional DDSFor a set of coupled PDEs, the user can create the FDAs and DDS’s using a Maple forloop. Note that FD will check for consistency of the LHS and RHS of each elementof DDS as well as the consistency between all the elements. It will raise errors if thefinite difference expression on the RHS does not have the same dimensionality asthe LHS. However at the moment FD does not check if the finite difference moleculeon the RHS fits into the domain specified on the LHS. The user need to be carefulwith the manual discretization of the equations, and to avoid out of range errors, itis best to use finite difference specifiers (FDS) discussed in Sec A.4.3.A.5.2 Imposing Outer Boundary ConditionsThe next step is to create the appropriate FDA expressions (the RHS expressionsin the DDS) that are compatible with specific boundary conditions and also arecreated under consideration that there are limitations on the allowed points in thevicinity of the boundary points. The case of fixing the value of the function orDirichlet boundary condition is quite simple and demonstrated in the example forheat equation. Often other types of boundary conditions appear in physical systems.One of particular interest is the “out-going” type or Neumann boundary condition.For a 1-D wave equation the out-going boundary condition is given by:∂tf(t, x) = −∂xf(t, x) (A.68)228A.5. Posing a PDE & Boundary Conditions Over a Discrete Domainfor the right side boundary i=Nx that corresponds to the approximate positive “in-finity” of the numerical domain x = +L∞. The out-going boundary condition atthe the left side of the numerical domain is given by:∂tf(t, x) = +∂xf(t, x) (A.69)for the point i=1 or x = −L∞. To implement such boundary conditions, oneneeds to use FDA expressions that can be evaluated at the point of boundary thatdoes not allow symmetric FD scheme. As described in Sec. A.4.3, this is achievedby changing the FDA scheme in FD using finite difference specifiers (FDS). Forexample, the following demonstrates an implementation of a mixed boundary forwave equation in which left boundary is fixed while the right boundary is outgoing.Note the change of FDA scheme using the FDS: fds backwardx.WaveEq := diff(f(t,x),t,t) - diff(f(t,x),x,x):WaveEqBdy := diff(f(t,x),t) + diff(f(t,x),x):WaveEqD := Gen_Sten(WaveEq):fds_backwardx:=table([ t=[-1,-1],x=[-1,0],y=[-1,-1],z=[-1,-1] ]):Update_FD_Table(2,fds_backwardx):WaveEqBdyD := Gen_Sten(WaveEqBdy):ddsWAVE:= [i=[1,1,1] = f(n+1,i) - myzero*x(i) ,i=[2,Nx-1,1] = WaveEqD,i=[Nx,Nx,1] = WaveEqBdyD];Example 10: Specifying outgoing/mixed boundary condition for wave equation229A.5. Posing a PDE & Boundary Conditions Over a Discrete DomainA.5.3 Periodic Boundary Condition: FD PeriodicAnother common boundary specification is periodic boundary condition (PBC). FDprovides a facility to implement PBCs by making a VDE periodic. The procedure:FD Periodic(exprd::VDE,{I=1/NI})takes an input, exprd, of type VDE and creates a periodic version of it. This locationis specified by the second argument, in which I is one of the indices (i,j,k) and theRHS is either 1 or NI where NI is one of the associated grid size (Nx,Ny,Nz). Forexample i=1 denotes that a periodic version of VDE is needed at the left boundaryand i=Nx denotes the same for the right boundary point. The replacements doneon FDA is illustrated in the following graph:........i=1i=NxF(i−1) (at i=1)  replaced wtih F(i+Nx−2)F(i+2) (at i=Nx) replaced with F(i−Nx+3)The following example, demonstrates the effect of the FD Periodic procedureon a VDE:> FD_table[x]:=[ [0], [-1,0,1] ,[-2,-1,0,1,2] ]:> A:= Gen_Sten(diff(f(x),x,x));f(i - 2) - 16 f(i - 1) + 30 f(i) - 16 f(i + 1) + f(i + 2)A := -1/12 ---------------------------------------------------------230A.5. Posing a PDE & Boundary Conditions Over a Discrete Domain2hx> FD_Periodic(A,{i=1});f(i - 3 + Nx) - 16 f(i - 2 + Nx) + 30 f(i) - 16 f(i + 1) + f(i + 2)-1/12 -------------------------------------------------------------------2hx> FD_Periodic(A,{i=Nx});f(i - 2) - 16 f(i - 1) + 30 f(i) - 16 f(i + 2 - Nx) + f(i + 3 - Nx)-1/12 -------------------------------------------------------------------2hxFinally, using this procedure, the implementation of a periodic DDS for wave equa-tion can be achieved as following:ddsWAVE_Periodic:= [{ i=[1,1,1] } = FD_Periodic(WaveEqD,{i=1}) ,{ i=[2,Nx-1,1] } = WaveEqD,{ i=[Nx,Nx,1] } = FD_Periodic(WaveEqD,{i=Nx})];Example 11: Implementation of a periodic boundary conditionin which WaveEqD is the same as Example 10.A.5.4 Implementing Ghost Cells for Odd and Even Functions:A FD Odd, A FD EvenThe boundaries of the numerical domain often correspond to the spatial infinity.However, a different coordinate system than Cartesian coordinate can be chosen,particularly to impose a certain symmetry. For example, one can work in a sphericalcoordinate and assume that the function’s spatial dependency is only of the form:f(t, x, y, z) = f(t, r) , r =√x2 + y2 + z2 . (A.70)231A.5. Posing a PDE & Boundary Conditions Over a Discrete DomainIn this case, the domain of the PDE (and the function f) is r ∈ (0,∞). Thepoint r = 0 is superficially a boundary of the numerical domain in this coordinatesystem, while in fact there is no physical boundary. These types of boundaries areoften referred as inner boundaries and usually are treated by imposing a specificbehaviour for the functions derived from the underlying symmetry.One common scenario that occurs in the point (or axis) of symmetry is thatfunctions (depending on what they represent: scalar, vector, component of a tensoretc) become even or odd. For example, a scalar function with spherical symmetry,ψ(t, r), is an even function at r = 0, i.e:ψ(t,−r) = ψ(t, r) (A.71)Note that here, −r is neither a physical location, nor is part of the numerical domain.However, one can simply consider the function along the x axis (where r = |x|) andthe refection symmetry x→ −x implies that the function is even in x. This means,equivalently, function is even in r if we consider an extension of it to r < 0 thatrepresents the value of the function at −x. More rigorously, the functions at thelimit of r→ 0 take one of the two forms:f(t, r) = C0(t) + C2(t)r2 + C4(t)r4 + . . . → function is even (A.72)f(t, r) = C1(t)r + C3(t)r3 + C5(t)r5 + . . . → function is odd (A.73)where the first case is for “scalar” functions and the second case is for “vector”functions.This property of the functions at inner boundaries allows a discretization tech-nique known as “ghost cells” in finite difference method. For example, consider thesecond and first derivative of the function with 4’th order accuracy:232A.5. Posing a PDE & Boundary Conditions Over a Discrete Domain# FD Updated to 4’th order before...> A:=Gen_Sten(diff(f(x),x,x));f(i - 2) - 16 f(i - 1) + 30 f(i) - 16 f(i + 1) + f(i + 2)A := -1/12 ---------------------------------------------------------2hx> B:=Gen_Sten(diff(f(x),x));f(i - 2) - 8 f(i - 1) + 8 f(i + 1) - f(i + 2)B := 1/12 ---------------------------------------------hxObviously, these terms cannot be used at i = 1, the left boundary point, and alsothe point next to it i = 2. However, if we impose the even or odd behaviour onthe function, the values f(i-1) and f(i-2) are known from the symmetry. Forexample, assuming that i is the point of symmetry, i.e the FDA will be used at i=1,then for an even function: f(i-1) = f(i+1). This condition is illustrated in thefollowing diagram (a). Similarly, for an odd function we have: f(i-2) = -f(i+2).Consider another example where f is odd, and the FDA will be used at i=2, thepoint next to the inner boundary point. The out-of-bound term in the FDA in thiscase is only f(i-2) and from the symmetry we must have: f(i-2) = -f(i). Thefollowing diagram (b) clarifies this condition:233A.5. Posing a PDE & Boundary Conditions Over a Discrete Domaini+1ii+2i−1 i+3i+1i−2 FF F F=(a):(b): F FF F ii−1i−2i−3 i+1= FiFi−2 −= i−1F FOne standard method to implement this symmetry is to actually extend thenumerical domain to have extra points outside the physical domain. These points,namely ghost cells, are updated via the symmetry, and allow FDA operations at theboundary point.FD provides a tool equivalent to the ghost cell technique. One can directlymanipulate the FDA expression according to the symmetry such that the out ofbound terms are replaced appropriately and the FDA can be used at the boundarypoints. FD provides two procedures to perform this task:A FD Even(exprd::VDE,coord,set of even funcs,symm loc,"forward/backward")A FD Odd(exprd::VDE,coord,set of odd funcs,symm loc,"forward/backward")where exprd is an FDA expression of type VDE, coord is the name of the coordinatewhich we are imposing the symmetry on (one of the (x,y,z)), the two variablesset of even funcs and set of odd funcs are of type set and include the nameof the functions that are even and odd respectively. symm loc is an integer that234A.5. Posing a PDE & Boundary Conditions Over a Discrete Domaindetermines the location of the inner boundary relative to the point where FDAwill be evaluated. For example the diagram (a) above corresponds to: symm loc= 0 and the diagram (b) can be imposed by setting: symm loc = -1. The lastargument is of type string, and determines if the replacement should be ”forward” –when the smaller index values are the out-of-bound ones and must be replaced – or”backward”, i.e the larger index values are the out-of-bound and require replacementwith indexed terms inside the physical domain. Normally, if the inner boundary ischosen to be the index i = 1 or ( j = 1, k = 1 in higher dimensions), these procedureswill only be use in ”forward” mode.The following demonstrates the usage of these two procedures and their output:> A:=Gen_Sten(diff(f(x),x,x));f(i - 2) - 16 f(i - 1) + 30 f(i) - 16 f(i + 1) + f(i + 2)A := -1/12 ---------------------------------------------------------2hx# Diagram (a) above:> A_FD_Even(A,x,{f},0,"forward");2 f(i + 2) - 32 f(i + 1) + 30 f(i)-1/12 ----------------------------------2hx# Diagram (b) above:> A_FD_Even(A,x,{f},-1,"forward");31 f(i) - 16 f(i - 1) - 16 f(i + 1) + f(i + 2)-1/12 ----------------------------------------------2hx> B:= Gen_Sten(diff(f(x),x));f(i - 2) - 8 f(i - 1) + 8 f(i + 1) - f(i + 2)B := 1/12 ---------------------------------------------hx235A.6. Solving a PDEs> A_FD_Odd(B,x,{f},0,"forward");-2 f(i + 2) + 16 f(i + 1)1/12 -------------------------hx> A_FD_Even(B,x,{f},0,"forward");0Example 12: Imposing even and odd symmetry at inner boundary pointNote that in the last execution, the result is identical to zero since the first derivativeof an even function is zero at the point of symmetry. We also note that if the FDAinvolves several functions of mixed even and odd type, both of the routines need tobe applied consecutively to the FDA to achieve a proper discretized version, usableat the inner boundary point.A.6 Solving a PDEsThis section demonstrates how to incorporate all of FD’s procedures and structuresto solve a PDE.A.6.1 Creating Initializer Routines: Gen Eval CodeThe first step is to create routines that initialize the function f(t = 0, x, y, z). If thisinitialization has an explicit function form depending on the coordinate and can beevaluated on every point of the numerical grid (x(i),y(j),z(k)) then it can besimply created using the procedure:Gen Eval Code(expr,input="c*/d",proc name="my init proc");where expr is either a continuous expression (setting input="c", this is the defaultsetting) or it is a VDE (by setting input="d"). The next option proc name is the236A.6. Solving a PDEsname of the Fortran procedure we want to create and it denotes both the name ofthe file (without the suffix .f) and the name of the procedure.For example, consider the case where we want to set the initial profile of thewave package to a Gaussian function:f(t = 0, x, y) = A exp(−(x− xc)2δ2x− (y − yc)2δ2y)(A.74)the following FD code performs the desired task:> read "../FD.mpl": MFD():Warning, grid_functions is not assignedFD table updated, see the content using SFDT() command> grid_functions:={f}:> init_f:=A*exp( -(x-xc)^2/delx^2 - (y-yc)^2/dely^2 ):> Gen_Eval_Code(init_f,input="c",proc_name="init_to_gauss");Fortran Code is written to init_to_gauss.fC header is written to init_to_gauss.hC call is written to init_to_gauss_callExample 13: Creating Initializer Fortran routinesSimilar to the very first example of creating IRE routines, all of FD’s code generatorroutines create 3 files, X.f, X.h and X call, where the Fortran file X.f is the bodyof the Fortran procedure that performs the desired task. All of the proceduresgenerated by FD have a last argument named res. For example the routine createdby the execution above, init to gauss.f has the following header:subroutine init_to_gauss(x,y,Nx,Ny,A,delx,dely,xc,yc, res)in which the highlighted res is the pointer to the vector that stores the returnedvalue of the procedure. In this case it is the function. Therefore user should passin the pointer that stores f at initial time in the driver code. The header file .h isa wrapper that can be included in a C driver program to use this routine. In theexample above, the content of the file, init to gauss.h is:void init_to_gauss_(double *x,double *y,int *Nx,int *Ny,double *A,double *delx,double *dely,double *xc,double *yc,double *res);237A.6. Solving a PDEsand finally the X call files are typical C calls that can be copied to the C driverand after changing the last argument res to the appropriate pointer, can be used tocall the Fortran routine. In the example above, the content of init to gauss callis:init_to_gauss_(x,y,&Nx,&Ny,&A,&delx,&dely,&xc,&yc,res);We note that if the expression that is passed to the procedure contains deriva-tives, (or FDA expressions in discrete form) then this procedure only evaluates/ini-tializes the expression at the points where the evaluation is possible i. e. allowed bythe size of the finite difference molecule (FDM). This usually results in a Fortranroutine that ignores the evaluation of the function on the boundary points (and per-haps in its vicinity depending on how large the resulting FDM is). If the evaluationis required at the boundary points, then the procedure described in the next sectionshould be used.A.6.2 Point-wise Evaluator Routines with DDS: A Gen Eval CodeIf the initialization is needed to vary at different portions of the discrete domain,the Fortran routine can be generated using the “evaluator” routine generator:A Gen Eval Code(dds:DDS,input="c*/d",proc name="my eval proc");where the only difference between this procedure and Gen Eval Code is in the firstargument, where this procedure accepts DDS type to allow specific calculations atdifferent parts of the domain. This procedure can also be used to evaluate a specificfunction that depends on the primary dynamical fields and their derivatives. It canalso be used to evaluate the point-wise residuals of PDEs if needed.For example, consider the problem where we want to evaluate the function f thatis the Laplacian of the function φ in cylindrical coordinate with axial symmetry:f(ρ, z) = ∇2φ = 1ρ∂ρ(ρ∂ρφ(ρ, z)) + ∂2zφ(ρ, z) = ∂2ρφ+∂ρφρ+ ∂2zφ (A.75)238A.6. Solving a PDEsNote that we would like to evaluate this Laplacian value on the axis ρ = 0 (which isan inner boundary) as well as the interior points. To do so, we need to deal with theirregular term 1ρ and also impose an inner boundary condition at ρ = 0. As discussedpreviously, these types of inner boundary conditions are dealt by looking into thebehaviour of the function at the limit of approaching the boundary, here: ρ → 0.We know that the function φ is a scalar and its first derivative ∂ρφ approaches zeroon the axis as O(ρ). Using the L’Hospital’s rule:limρ→0∂ρφρ=∂2φ∂ρ2(A.76)Therefore, two versions of the expression are used to evaluate f :f = ∇2φ =∂2ρφ+∂ρφρ + ∂2zφ if ρ 6= 02∂2ρφ+ ∂2zφ if ρ = 0(A.77)In addition, the evaluation of the derivative as an FDA at ρ = requires implemen-tation of boundary condition as described in Sec. A.5.4. Here the inner boundarycondition is created using the fact that φ is an even function in ρ. The followingexample demonstrates all of the steps described to achieve this evaluation:read "../FD.mpl": CFD(): MFD():grid_functions:={phi}:Laplace_Interiour:= diff(phi(x,z),x,x) + diff(phi(x,z),x)/x + diff(phi(x,z),z,z):Laplace_Boundary_D:= Gen_Sten(2*diff(phi(x,z),x,x) + diff(phi(x,z),z,z)):dds_2Dlaplace:= [{ i=[2,Nx-1,1] , k=[2,Nz-1,1] } = Gen_Sten(Laplace_Interiour),{ i=[1,1,1], k = [2,Nz-1,1] } = A_FD_Even(Laplace_Boundary_D,x,{phi},0,"forward"),{ i=[Nx,Nx,1] , k=[1,Nz,1] } = myzero*x(i)*z(k),{ i=[1,Nx,1] , k =[1,1,1] } = myzero*x(i)*z(k),{ i=[1,Nx,1] , k =[Nz,Nz,1] } = myzero*x(i)*z(k)]:239A.6. Solving a PDEsA_Gen_Eval_Code(dds_2Dlaplace,input="c",proc_name="eval_laplace");Fortran Code is written to eval_laplace.fC header is written to eval_laplace.hC call is written to eval_laplace_callExample 14: Point-wise Evaluator Routine Generator Using a DDSA.6.3 Creating IRE Testing Routines: Gen Res CodeIf the function that needs to be evaluated is indeed a residual, i.e. expected tobe zero in the continuum limit, then often the user is interested in monitoring thel2-norm of this residual. FD provides a procedure that creates a Fortran routine forsuch an evaluation:Gen Res Code(expr,input="c*/d",proc name="my res proc");where expr can be a PDE residual in a continuous form or a VDE. The only differ-ence between this procedure and Gen Eval Code is that the Fortran routine gener-ated here will perform a l2-norm (root mean square to be specific) on the functionand returns a single real number. This routine can be used as a fast prototyping toolto create Independent Residual Evaluator routines. The following demonstrates anexample of creating IRE for wave equation:read "../FD.mpl": Clean_FD(): Make_FD():grid_functions := {f}:WaveEq := diff(f(t,x),t,t) = diff(f(t,x),x,x):Gen_Res_Code(lhs(WaveEq)-rhs(WaveEq),input="c",proc_name="ire_wave");Example 15: Fast Prototyping IRE RoutinesA.6.4 Creating Piece-wise Residual Evaluator RoutinesSimilar to the generalization of Gen Eval Code to A Gen Eval Code such that theprocedure accepts a DDS such that the function can be evaluated on each portion240A.6. Solving a PDEsof the discrete domain, here A Gen Res Code extends the capability of previous pro-cedure Gen Res Code to evaluate the l2-norm of the residual that is specified by aDDS:A Gen Res Code(dds:DDS,input="d/c*",proc name="my res proc");This procedure is perhaps most useful to evaluate the norm of the residual of thePDE under study. The returned norm of the residual can be compared to a thresholdvalue to determine if the PDE is numerically solved after applying the solver rou-tine (or after certain number of iterations of the solver routines are applied). Notethat this routine can also be used as an IRE generator. Example 14 can be usedto demonstrate the use of this procedure, the difference is that the Fortran routinecreated by this procedure will return the l2-norm of the laplace equation, and there-fore can be useful if we are monitoring the norm of the residual and convergence ofour numerical solver.A.6.5 Creating Solver Routine: A Gen Solve CodeAs we discussed in Sec. A.2.2, for a given FDA of a PDE written in canonical form:PDE = L(f) = 0 = Lh(fh) (A.78)the solving process involves finding the unknowns fijk (for a boundary value prob-lem) or fn+1ijk for initial value problem using fnijk. As introduced in Sec. A.2.2, astandard approach for a nonlinear system is to use Newton-Gauss-Seidel iterativemethod. FD provides a procedure that generates routines which implement singleiteration of this method:A Gen Solve Code(dds:DDS,{solve for var},input="d/c*",proc name="my solver proc");where the first argument is of type DDS, and the second argument is a set of unknownsfor which the FDA must be solved. At the moment this set must contain only a single241A.6. Solving a PDEsterm, such as f(n+1,i,j,k) as the unknown. The created Fortran routine performsa single iteration of Newton-Gauss-Seidel and returns the “updated” function in thelast argument, namely res which shall be adjusted by the user. This completesall the necessary tools to create a set of solver routines for a PDE, and in thenext section we put together all of the features of FD discussed to demonstrate animplementation of a solver system for 1-D wave equation using an implicit scheme.This example also demonstrates the use of this solver procedure.Note on myzero ExpressionAs it has been seen at several points in this document, the user needs to im-plement constant functions or residual equations by adding a trivial VDE such asmyzero*x(i)*y(j). This is due to the fact that FD uses VDE’s to figure out thedimensionality and dependencies of the PDEs, therefore if a single expression suchas a constant number is given to FD’s discretization routines, it has no way of find-ing the dimensionality of the problem. In particular, the common scenario thatthe use of myzero is essential is when in the equation that needs to be solved thesolution simplifies to a single constant or zero. For instance, in Example 10 we areimposing fixed boundary condition f = 0 at x = 0, therefore the residual of theequation (LHS of equation in canonical form: L(f) = 0) is simply: f . However, theimplementation of this residual has to be f −myzero ∗ x, since if f is passed in asthe residual, the solver VDE simplifies to 0, which has no valid dependency on anydiscrete index, {i,j,k,n}, to be understood by FD.A.6.6 Communicating with Parallel Computing InfrastructureHere we present a simple communication method with a parallelization infrastruc-ture (FD adopts PAMR’s [33] standard). To achieve this goal a vector of integerflags, phys bdy is passed to the solver/evaluator routines in which the value 1 de-notes that the boundary is a real physical boundary, therefore the boundary condi-tion should be imposed, and the value 0 denotes that it is a boundary between CPUs242A.6. Solving a PDEsand usually no calculation is required as the parallel frameworks often implementbetween CPU ghost cells for the distributed sub-domains. These flags are invokedby setting the variable b to their associated names xmin,xmax,... as noted in ta-ble .(A.62) The following example demonstrates a DDS that implements boundaryflags:ddsfWave := [{ i=[2,Nx-1,1] , j=[2,Ny-1,1] } = PDEWave_D,{ i=[1,1,1] , j=[1,Ny,1] , b=xmin } = f(n+1,i,j) - myzero*x(i)*y(j),{ i=[Nx,Nx,1] , j=[1,Ny,1] , b=xmax } = f(n+1,i,j) - myzero*x(i)*y(j),{ i=[1,Nx,1] , j=[1,1,1] , b=ymin } = f(n+1,i,j) - myzero*x(i)*y(j),{ i=[1,Nx,1] , j=[Ny,Ny,1] , b=ymax } = f(n+1,i,j) - myzero*x(i)*y(j)];We encourage the reader to look into the Fortran files that are created using thistype of DDS to inspect how the phys bdy flags are positioned in the file.The tutorial: FD/tutorials/wave2d pamr fixed boundary in the distributionpackage is an implementation of a parallel 2 dimensional wave equation solver.A.6.7 Example: Crank-Nicolson Implementation of WaveEquationWe complete this section by combining all of the tools we discussed to a single Maplescript that creates a solver routine, residual evaluator and an independent residualevaluator as well as an initializer routine for the 1 dimensional wave equation. Thewave equation is given by:∂2t f(t, x) = ∂2xf(t, x) , (A.79)243A.6. Solving a PDEsand can be reduced to a first order system by defining ft as:ft(t, x) ≡ ∂tf(t, x) (A.80)⇒∂tf(t, x) = ft(t, x) (A.81)∂tft(t, x) = ∂2xf(t, x) (A.82)Here we assume periodic boundary conditions Note that the example, first imple-ments an IRE for the system using the original form of the wave equation and FD’sdefault second order leap-frog scheme. After that, the FD scheme is updated toforwards in time, and by virtue of time averaging we achieve second order accuracy.It also demonstrate how to create initializer routines as well as residual evaluatorroutines to measure how accurate the PDE is solved.read "../FD.mpl": Clean_FD(); Make_FD();grid_functions := {f,f_t};eq1 := diff(f(t,x),t) = f_t(t,x);eq2 := diff(f_t(t,x),t) = diff(f(t,x),x,x);eq3 := diff(f(t,x),t,t) = diff(f(t,x),x,x);Gen_Res_Code(lhs(eq3)-rhs(eq3),input="c",proc_name="ire_f");FD_table[t] := [[0],[0,1]];AVGT := a -> ( FD( a,[ [1],[0] ]) + FD( a,[ [0],[0] ]) )/2;eq1_D := Gen_Sten(lhs(eq1)) - AVGT(Gen_Sten(rhs(eq1)));eq2_D := Gen_Sten(lhs(eq2)) - AVGT(Gen_Sten(rhs(eq2)));init_f:=A*exp(-(x-x0)^2/delx^2);init_f_t:=idsignum*diff(init_f,x);Gen_Eval_Code(init_f,input="c",proc_name="init_f");Gen_Eval_Code(init_f_t,input="c",proc_name="init_f_t");244A.7. List of Abbreviationsdss_f:= [{ i=[1,1,1] } = FD_Periodic(eq1_D,{i=1}) ,{ i=[2,Nx-1,1] } = eq1_D,{ i=[Nx,Nx,1] } = FD_Periodic(eq1_D,{i=Nx})];dss_f_t:= [{ i=[1,1,1] } = FD_Periodic(eq2_D,{i=1}) ,{ i=[2,Nx-1,1] } = eq2_D,{ i=[Nx,Nx,1] } = FD_Periodic(eq2_D,{i=Nx})];A_Gen_Res_Code(dss_f,input="d",proc_name="res_f",is_periodic=true);A_Gen_Res_Code(dss_f_t,input="d",proc_name="res_f_t",is_periodic=true);A_Gen_Solve_Code(dss_f,{f(n+1,i)},input="d",proc_name="u_f",is_periodic=true);A_Gen_Solve_Code(dss_f_t,{f_t(n+1,i)},input="d",proc_name="u_f_t",is_periodic=true);Example 15: Implementation of Crank-Nicolson Scheme to Solve 1D WaveEq. (A.79)Note that several other complete examples are included in FD’s distributionpackage in the directory tutorials, including: 2D wave equation in parallel, non-linear mixed boundary 1D wave equation, heat equation, and 2D wave equation incylindrical coordinate with axial symmetry. All of the examples in this manual arealso included in the distribution under examples directory.Also see: http://laplace.phas.ubc.ca/People/arman/FD doc/tutorials.htmlfor detailed tutorials on how to use FD.A.7 List of AbbreviationsBVE: Boundary Value ProblemDDS: Discrete Domain SpecifierFD: Finite Difference, also the name of the toolkit245A.7. List of AbbreviationsFDA: Finite Difference ApproximationFDE: Finite Difference EquationFDM: Finite Difference MoleculeFDS: Finite Difference SpecifierIVE: Initial Value ProblemLHS: Left Hand SideODE: Ordinary Differential EquationPBC: Periodic Boundary Condition PDE: Partial Differential EquationRHS: Right Hand SideVCE: Valid Continuous ExpressionVDE: Valid Discrete Expression246


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