Treating instabilities in a hyperbolic formulation of Einstein's equations

Details Treating instabilities in a hyperbolic formulation of Einstein's equations Treating instabilities in a hyperbolic formulation of Einstein's equations

1998-07-14

arxiv.org/abs/gr-qc/9807029v1

Phys.Rev. D58 (1998) 044020

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

13 pages, 9 figures. To appear in Phys. Rev. D

Scientific paper

10.1103/PhysRevD.58.044020

We have recently constructed a numerical code that evolves a spherically symmetric spacetime using a hyperbolic formulation of Einstein's equations. For the case of a Schwarzschild black hole, this code works well at early times, but quickly becomes inaccurate on a time scale of 10-100 M, where M is the mass of the hole. We present an analytic method that facilitates the detection of instabilities. Using this method, we identify a term in the evolution equations that leads to a rapidly-growing mode in the solution. After eliminating this term from the evolution equations by means of algebraic constraints, we can achieve free evolution for times exceeding 10000M. We discuss the implications for three-dimensional simulations.

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