The discrete energy method in numerical relativity: Towards long-term stability

TL;DR

This paper explores the application of the energy method to develop stable finite difference schemes in numerical relativity, addressing artificial instabilities and proposing solutions to improve long-term simulation stability.

Contribution

It extends the energy method analysis to discrete systems in numerical relativity, identifying causes of artificial instabilities and suggesting ways to mitigate them.

Findings

Convergent schemes are achievable for test problems.

Artificial instabilities can arise from discrete derivative approximations.

Partial mitigation of growth modes is possible through proposed techniques.

Abstract

The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete system can be used to construct stable finite difference equations for these problems at the linear level. In this paper we apply these techniques to some test problems commonly used in numerical relativity and observe that while we obtain convergent schemes, fast growing modes, or ``artificial instabilities,'' contaminate the solution. We find that these growing modes can partially arise from the lack of a Leibnitz rule for discrete derivatives and discuss ways to limit this spurious growth.