Finite difference schemes for second order systems describing black holes
In the harmonic description of general relativity, the principal part of Einstein’s equations reduces to 10 curved space wave equations for the components 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.
Motamed, M., Babiuc, M., Szilágyi, B., Kreiss, H. O., & Winicour, J. (2006). Finite difference schemes for second order systems describing black holes. Physical Review D, 73(12), 124008.