Finite differencing second order systems describing black hole spacetimes

TL;DR

This paper investigates the stability of second order finite differencing schemes for Einstein's equations in black hole spacetime simulations, identifying issues with standard discretizations and proposing effective first order reduction methods.

Contribution

It demonstrates the instability of standard second order discretizations with large shift parameters and advocates for first order reduction techniques for stable numerical relativity simulations.

Findings

Standard second order schemes become unstable with large shift parameters.

First order reduction based on ADM variables improves stability.

Discretization issues are critical in black hole spacetime simulations.

Abstract

Keeping Einstein's equations in second order form can be appealing for computational efficiency, because of the reduced number of variables and constraints. Stability issues emerge, however, which are not present in first order formulations. We show that a standard discretization of the second order ``shifted'' wave equation leads to an unstable semi-discrete scheme if the shift parameter is too large. This implies that discretizations obtained using integrators such as Runge-Kutta, Crank-Nicholson, leap-frog are unstable for any fixed value of the Courant factor. We argue that this situation arises in numerical relativity, particularly in simulations of spacetimes containing black holes, and discuss several ways of circumventing this problem. We find that the first order reduction in time based on ``ADM'' type variables is very effective.