  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.
