Testing the Accuracy and Stability of Spectral Methods in Numerical Relativity

TL;DR

This paper evaluates the accuracy and stability of spectral methods in numerical relativity, demonstrating exponential convergence and analyzing error growth, with implications for high-amplitude gauge wave simulations.

Contribution

It adapts standard tests for spectral methods in numerical relativity and derives a general error growth expression, showing spectral methods' high accuracy and stability limits.

Findings

Spectral code exhibits exponential convergence limited only by roundoff error.

Errors grow as predicted by the derived error expression in linear tests.

Most simulations are stable except for high amplitude gauge waves with nontrivial shift.

Abstract

The accuracy and stability of the Caltech-Cornell pseudospectral code is evaluated using the KST representation of the Einstein evolution equations. The basic "Mexico City Tests" widely adopted by the numerical relativity community are adapted here for codes based on spectral methods. Exponential convergence of the spectral code is established, apparently limited only by numerical roundoff error. A general expression for the growth of errors due to finite machine precision is derived, and it is shown that this limit is achieved here for the linear plane-wave test. All of these tests are found to be stable, except for simulations of high amplitude gauge waves with nontrivial shift.