On the Numerical Stability of the Einstein Equations

TL;DR

This paper analyzes the numerical stability of different discretizations of Einstein equations, revealing that the conformal-traceless formulation is stable under certain conditions, unlike the standard ADM formulation.

Contribution

It provides a von Neumann stability analysis comparing ADM and CT formulations, highlighting stability conditions for nonlinear plane wave solutions.

Findings

ADM formulation is unconditionally unstable for nonlinear plane waves.

CT formulation remains stable for certain timestep ratios.

Stability condition for flat space is t/z  1.

Abstract

We perform a von Neumann stability analysis on a common discretization of the Einstein equations. The analysis is performed on two formulations of the Einstein equations, namely, the standard ADM formulation and the conformal-traceless (CT) formulation. The eigenvalues of the amplification matrix are computed for flat space as well as for a highly nonlinear plane wave exact solution. We find that for the flat space initial data, the condition for stability is simply $\frac {\Delta t}{\Delta z} \leq 1$. However, a von Neumann analysis for highly nonlinear plane wave initial data shows that the standard ADM formulation is unconditionally unstable, while the conformal-traceless (CT) formulation is stable for $0.25 \leq \frac {\Delta t}{\Delta z} < 1$.