Evolving a puncture black hole with fixed mesh refinement

TL;DR

This work addresses how fixed mesh refinement interfaces affect numerical relativity simulations of dynamical black holes. It adopts the BSSN formulation with puncture data and implements a third-order guard cell filling scheme within Paramesh to maintain second-order convergence near refinement boundaries while expanding the computational domain beyond $100M$. Through geodesic slicing and a variant of 1+$\,$log slicing, it demonstrates robust convergence in strong-field regions and identifies gauge-dependent limits, including a high-frequency origin-related noise under certain conditions. The results support the viability of FMR for long, high-resolution simulations and efficient gravitational waveform extraction, with future work extending to non-zero shift gauges.

Abstract

We present an algorithm for treating mesh refinement interfaces in numerical relativity. We detail the behavior of the solution near such interfaces located in the strong field regions of dynamical black hole spacetimes, with particular attention to the convergence properties of the simulations. In our applications of this technique to the evolution of puncture initial data with vanishing shift, we demonstrate that it is possible to simultaneously maintain second order convergence near the puncture and extend the outer boundary beyond 100M, thereby approaching the asymptotically flat region in which boundary condition problems are less difficult and wave extraction is meaningful.

Figures (12)

• Figure 1: Guard cell filling in two spatial dimensions. In these pictures, the thick vertical line represents a refinement boundary separating fine and coarse grid regions. The picture on the left shows the first step, in which one of the parent grid cells (grey square) is filled using quadratic interpolation across nine interior fine grid cells (black circles). The other parent grid cells are filled using corresponding stencils of nine interior fine grid cells. (The asymmetry in the left panel is drawn with the assumption that the fine block's center is toward the top-left of the panel.) The picture on the right shows the second step in which two fine grid guard cells (grey circles) are filled using quadratic interpolation across nine parent grid values (squares). These parent grid values include one layer of guard cells (black squares) obtained from the coarse grid region to the right of the interface, and two layers of interior cells (grey squares). The final step in guard cell filling (not shown in this figure) is to use "derivative matching" to fill the guard cells for the coarse grid.
• Figure 2: Evolution of the conformal metric component $\tilde{\gamma}_{xx}$, for a geodesically sliced puncture, shown at $t=0.5$, 1.0, 1.5, 2.0, and $2.5 M$.
• Figure 3: Convergence of the errors (numerical values minus analytic values) in $\tilde{\gamma}_{xx}$, $\tilde{A}_{xx}$, the Hamiltonian constraint $H$, and the momentum constraint $P^{x}$, for a geodesically sliced puncture along the $x$-axis, all at the time $t=2.5 M$. The solid line shows the errors for the highest resolution run. The errors for the medium resolution run (dashed line) and the lowest resolution run (dotted line) have been divided by factors of 4 and 16, respectively, to demonstrate second order convergence. Note that the full domain of the simulation extends to $128M$.
• Figure 4: Close-up of the convergence of the error (numerical value minus analytic value) in $\tilde{\gamma}_{xx}$, for a geodesically sliced puncture, at $t = 2.5M$, in the vicinity of the refinement boundary at $x = 2M$. The errors for the high resolution run are shown by the solid line; the errors for the medium (dashed line) and low (dotted line) resolution runs have been divided by factors of 4 and 16, respectively. In this plot, we also show the location of the data points, including guardcells, using filled circles.
• Figure 5: A view of the Hamiltonian constraint in the neighborhood of the FMR interface at $x=2M$. The errors for the high resolution run are shown by the solid line; the errors for the medium (dashed line) and low (dotted line) resolution runs have been divided by factors of 4 and 16, respectively. As discussed in the text, data points nearest to the interface converge at one order lower than in the rest of the domain.