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.

Figure 1

Convergence plot for the Hamiltonian constraint error $H$ at time $t=4M$. There is a single puncture black hole centered at the origin, and refinement boundaries at $|x_i|=1M$, $2M$, and $4M$. The finest grid spacing $h_f$ for each simulation is indicated in the figure, and the highest resolution is multiplied by a factor of 4 while the lowest resolution is divided by a factor of 4. For second-order errors the curves should superpose in the limit $h\to 0$.

Figure 2

Convergence plot for the error in $\phi$. There is a refinement boundary at $x=50$. The coarse grid spacing $h$ of each simulation is indicated in the figure, and each curve has been multiplied by a respective factor as appropriate to demonstrate second-order convergence. The convergence is demonstrable only for resolutions of $h\lesssim 5/128$.

Figure 3

Numerical error in $\phi$, immediately after a pulse incident on the refinement interface at $x=50$ has passed through, generating a pulse of transmitted error and a contracted pulse of reflected error. The error is second-order convergent and in agreement with the analytic prediction of the numerical error, as indicated.

Figure 4

Comparison of the error in $\phi$ reflected from the refinement boundary at $x=50$ in the case of a second-order conventional stencil and a second-order MAD stencil.

Figure 5

Comparison of the Hamiltonian constraint $H$ in the case of second-order accurate, conventional differencing stencils and second-order accurate, MAD stencils, as well as a demonstration of convergence in the latter case. There is a single puncture black hole centered at the origin, and refinement boundaries at $|x_i|=1M$, $2M$, and $4M$.