UBC Theses and Dissertations
Numerical relativity in black hole spacetimes Thornburg, Jonathan
This thesis is concerned with the development of better techniques for the 3 + 1 numerical relativity study of black hole spacetimes. The main result of this thesis is the development of a new technique for avoiding singularities in such spacetimes. In this technique the slices are allowed to penetrate the black hole, but only the region of spacetime outside the apparent horizon is numerically evolved. This allows the slicing to be chosen to avoid the "grid stretching" problems commonly encountered when freezing slicings are used. To implement this scheme, we have developed a robust and efficient apparent-horizon-finding algorithm. We use this at each time step during a numerical evolution to monitor the apparent horizon’s position; we then dynamically adjust the region of spacetime excluded from the numerical evolution so this region tracks the apparent horizon's motion. In this thesis we use coordinates in which all components of the metric and other 3 + 1 field tensors are (generally) nonzero. This makes the 3 + 1 equations very complex, so we have developed a prototype "PDE Compiler" to automatically finite difference them and generate the required code. This automation of the finite differencing process allows us to work with and think about the 3 + 1equations almost entirely at the tensor-differential-operator level. We have developed a new initial data solver, which numerically solves the full 4-vector York equations on slices which are generally not maximal and not 3-conformally-flat. Our numerical methods are based on 4th order finite differencing, using the method of lines for hyperbolic PDEs. To study these and our black hole exclusion technique in a simple setting, we have made a series of model problem studies using a 1-dimensional flat-space scalar wave equation. These have been very successful, yielding a stable and highly accurate finite differencing scheme. To test these techniques in a more realistic setting, we have written a prototype numerical relativity code to simulate the time evolution of axisymmetric (single) black hole spacetimes. At present, this code suffers from severe finite differencing instabilities. We have identified potential causes for some of these instabilities, but we have not yet resolved them.
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