Summation by parts and dissipation for domains with excised regions

TL;DR

This paper develops finite difference methods with summation by parts properties for hyperbolic equations in complex domains, enabling stable simulations of black hole spacetimes with excised regions.

Contribution

It introduces dissipative and difference operators satisfying summation by parts in domains with multiple excised regions, aiding stability analysis.

Findings

Operators satisfy summation by parts in complex domains

Energy estimates derived for stability

Applicable to black hole spacetime simulations

Abstract

We discuss finite difference techniques for hyperbolic equations in non-trivial domains, as those that arise when simulating black hole spacetimes. In particular, we construct dissipative and difference operators that satisfy the {\it summation by parts} property in domains with excised multiple cubic regions. This property can be used to derive semi-discrete energy estimates for the associated initial-boundary value problem which in turn can be used to prove numerical stability.