Testing numerical relativity with the shifted gauge wave

TL;DR

This paper introduces a shifted gauge wave test for numerical relativity, revealing an exponential instability caused by shift, and discusses techniques to suppress it, aiding black hole simulations.

Contribution

It presents a new shifted gauge wave test with boundary conditions that expose instabilities, and proposes methods to suppress these in harmonic evolution codes.

Findings

Shift introduces exponential growth in the gauge wave test.

Certain techniques can suppress instabilities caused by shift.

The methods are applicable to black hole simulations.

Abstract

Computational methods are essential to provide waveforms from coalescing black holes, which are expected to produce strong signals for the gravitational wave observatories being developed. Although partial simulations of the coalescence have been reported, scientifically useful waveforms have so far not been delivered. The goal of the AppleswithApples (AwA) Alliance is to design, coordinate and document standardized code tests for comparing numerical relativity codes. The first round of AwA tests have now being completed and the results are being analyzed. These initial tests are based upon periodic boundary conditions designed to isolate performance of the main evolution code. Here we describe and carry out an additional test with periodic boundary conditions which deals with an essential feature of the black hole excision problem, namely a non-vanishing shift. The test is a shifted version of the existing AwA gauge wave test. We show how a shift introduces an exponentially growing instability which violates the constraints of a standard harmonic formulation of Einstein's equations. We analyze the Cauchy problem in a harmonic gauge and discuss particular options for suppressing instabilities in the gauge wave tests. We implement these techniques in a finite difference evolution algorithm and present test results. Although our application here is limited to a model problem, the techniques should benefit the simulation of black holes using harmonic evolution codes.