Finite difference schemes for second order systems describing black holes

TL;DR

This paper develops and analyzes stable finite difference algorithms for second order wave equations in black hole spacetimes, demonstrating their effectiveness through numerical experiments.

Contribution

It provides new stability theorems for evolution-boundary algorithms applied to second order Einstein equations in black hole models.

Findings

Algorithms are stable and convergent in black hole excision scenarios.

Theoretical stability results are validated through numerical experiments.

Performance comparison shows effectiveness of proposed schemes.

Abstract

In the harmonic description of general relativity, the principle part of Einstein's equations reduces to 10 curved space wave equations for the componenets of the space-time metric. We present theorems regarding the stability of several evolution-boundary algorithms for such equations when treated in second order differential form. The theorems apply to a model black hole space-time consisting of a spacelike inner boundary excising the singularity, a timelike outer boundary and a horizon in between. These algorithms are implemented as stable, convergent numerical codes and their performance is compared in a 2-dimensional excision problem.