Treating instabilities in a hyperbolic formulation of Einstein's equations

TL;DR

This paper identifies and eliminates a term causing instabilities in a hyperbolic formulation of Einstein's equations, significantly improving the stability and duration of numerical simulations of black holes.

Contribution

The authors develop an analytic method to detect instabilities and modify the evolution equations, enabling stable, long-term simulations of black hole spacetimes.

Findings

Instabilities arise from a specific term in the evolution equations.

Eliminating this term extends stable evolution to over 10,000 M.

The method has implications for 3D numerical relativity simulations.

Abstract

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.