Finite differencing second order systems describing black hole spacetimes

Details Finite differencing second order systems describing black hole spacetimes Finite differencing second order systems describing black hole spacetimes

2004-10-14

arxiv.org/abs/gr-qc/0410062v1

Phys.Rev. D71 (2005) 027501

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Scientific paper

10.1103/PhysRevD.71.027501

Keeping Einstein's equations in second order form can be appealing for computational efficiency, because of the reduced number of variables and constraints. Stability issues emerge, however, which are not present in first order formulations. We show that a standard discretization of the second order ``shifted'' wave equation leads to an unstable semi-discrete scheme if the shift parameter is too large. This implies that discretizations obtained using integrators such as Runge-Kutta, Crank-Nicholson, leap-frog are unstable for any fixed value of the Courant factor. We argue that this situation arises in numerical relativity, particularly in simulations of spacetimes containing black holes, and discuss several ways of circumventing this problem. We find that the first order reduction in time based on ``ADM'' type variables is very effective.

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