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On the Numerical Stability of the Einstein Equations

Mark Miller
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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 $\frac {Δt}{Δz} \leq 1$. 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 $0.25 \leq \frac {Δt}{Δz} < 1$.

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Mark Miller. 2000-08-08. On the Numerical Stability of the Einstein Equations. https://arxiv.org/abs/gr-qc/0008017

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