Numerical stability for finite difference approximations of Einstein's equations

TL;DR

This paper extends the concept of numerical stability to second order hyperbolic systems like Einstein's equations, providing conditions for stability, analyzing discretizations, and proposing testing methods.

Contribution

It introduces a stability analysis framework for second order hyperbolic systems, including Einstein's equations, and offers practical methods for stability testing.

Findings

Necessary and sufficient stability conditions derived

Standard discretizations can be unstable for well-posed problems

Courant limits are not always aligned with characteristic speeds

Abstract

We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in Numerical Relativity. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We prove stability for certain toy models and the linearized Nagy-Ortiz-Reula formulation of Einstein's equations. We also find that, unlike in the fully first order case, standard discretizations of some well-posed problems lead to unstable schemes and that the Courant limits are not always simply related to the characteristic speeds of the continuum problem. Finally, we propose methods for testing stability for second order in space hyperbolic systems.