Reducing reflections from mesh refinement interfaces in numerical relativity

TL;DR

This paper identifies and analyzes artificial reflections caused by mesh refinement interfaces in numerical relativity simulations, and proposes a finite differencing modification to eliminate this issue, improving simulation accuracy for gravitational wave data interpretation.

Contribution

It introduces mesh-adapted differencing (MAD), a novel finite differencing approach that removes interface reflections in numerical relativity simulations.

Findings

MAD effectively eliminates artificial reflections at refinement boundaries.

The analysis links slow convergence to mode speed differences in relativity.

The method improves the accuracy of black hole merger simulations.

Abstract

Full interpretation of data from gravitational wave observations will require accurate numerical simulations of source systems, particularly binary black hole mergers. A leading approach to improving accuracy in numerical relativity simulations of black hole systems is through fixed or adaptive mesh refinement techniques. We describe a manifestation of numerical interface truncation error which appears as slowly converging, artificial reflections from refinement boundaries in a broad class of mesh refinement implementations, potentially compromising the effectiveness of mesh refinement techniques for some numerical relativity applications if left untreated. We elucidate this numerical effect by presenting a model problem which exhibits the phenomenon, but which is simple enough that its numerical error can be understood analytically. Our analysis shows that the effect is caused by variations in finite differencing error generated across low and high resolution regions, and that its slow convergence is caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Lastly, we resolve the problem, presenting a class of finite differencing stencil modifications, termed mesh-adapted differencing (MAD), which eliminate this pathology in both our model problem and in numerical relativity examples.