On the Numerical Stability of the Einstein Equations

Details On the Numerical Stability of the Einstein Equations On the Numerical Stability of the Einstein Equations

2000-08-08

arxiv.org/abs/gr-qc/0008017v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

14 pages, 9 figures, submitted to Phys. Rev. D

Scientific paper

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 $0.25 \leq \frac {\Delta t}{\Delta z} < 1$.

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