Reducing reflections from mesh refinement interfaces in numerical relativity

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 significant, 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 associated difficulties in demonstrating convergence at modest resolutions are caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Last, to further verify our understanding of this problem, we present a class of finite differencing stencils of the same order of accuracy as the desired order of convergence, termed mesh-adapted differencing (MAD), which eliminate this pathology in both our model problem and in numerical relativity examples.