Convergence and stability in numerical relativity

TL;DR

This paper demonstrates that many numerical schemes used in Einstein's equations can be non-convergent due to instabilities, emphasizing the importance of proper convergence testing in numerical relativity.

Contribution

It explicitly shows how certain discretizations lead to non-convergent schemes in numerical relativity and proposes tests to detect such issues.

Findings

Many schemes amplify small errors without bound

Non-convergence can be missed by standard tests

Proposed methods improve detection of instabilities

Abstract

It is often the case in numerical relativity that schemes that are known to be convergent for well posed systems are used in evolutions of weakly hyperbolic (WH) formulations of Einstein's equations. Here we explicitly show that with several of the discretizations that have been used through out the years, this procedure leads to non-convergent schemes. That is, arbitrarily small initial errors are amplified without bound when resolution is increased, independently of the amount of numerical dissipation introduced. The lack of convergence introduced by this instability can be particularly subtle, in the sense that it can be missed by several convergence tests, especially in 3+1 dimensional codes. We propose tests and methods to analyze convergence that may help detect these situations.