Evolutions in 3D numerical relativity using fixed mesh refinement

TL;DR

The paper demonstrates fixed mesh refinement (FMR) in 3D numerical relativity using the Carpet driver within the Cactus framework, showing that FMR can reproduce the accuracy and stability of unigrid simulations at equivalent resolutions while significantly reducing computational cost. It introduces a robust infrastructure for non-adaptive yet fixed refinement hierarchies, including a buffer-zone technique to preserve convergence at grid interfaces and a novel initial-data evolution scheme enabling higher-order time interpolation from the initial slice. Through a suite of tests—wave propagation, a Gaussian pulse, 1/r data with excision, robust stability, a gauge wave, and a Schwarzschild black hole with excision—the authors demonstrate second-order convergence and close agreement with unigrid results, with notable efficiency gains (roughly 30% of resources) in refined runs. The work provides a practical, open-source path for the NR community to leverage FMR on legacy codes without substantial code rewrites, while outlining conditions (e.g., second-derivative systems with multi-step time integration) where buffer zones are essential.

Abstract

We present results of 3D numerical simulations using a finite difference code featuring fixed mesh refinement (FMR), in which a subset of the computational domain is refined in space and time. We apply this code to a series of test cases including a robust stability test, a nonlinear gauge wave and an excised Schwarzschild black hole in an evolving gauge. We find that the mesh refinement results are comparable in accuracy, stability and convergence to unigrid simulations with the same effective resolution. At the same time, the use of FMR reduces the computational resources needed to obtain a given accuracy. Particular care must be taken at the interfaces between coarse and fine grids to avoid a loss of convergence at higher resolutions, and we introduce the use of "buffer zones" as one resolution of this issue. We also introduce a new method for initial data generation, which enables higher-order interpolation in time even from the initial time slice. This FMR system, "Carpet", is a driver module in the freely available Cactus computational infrastructure, and is able to endow generic existing Cactus simulation modules ("thorns") with FMR with little or no extra effort.

Figures (15)

• Figure 1: Base level $G^0_0$ and two refined levels $G^1_j$ and $G^2_j$, showing the grid alignments and demonstrating proper nesting.
• Figure 2: Schematic for the prolongation scheme, in $1+1$ dimensions, for a two-grid hierarchy. The large filled (red) circles represent data on the coarse grid, and smaller filled (green) circles represent data on the fine grid. The arrows indicate interpolation of coarse grid data in space and time, necessary for the boundary conditions on the fine grid (explained in section \ref{['sec:boundary-prolongation']}).
• Figure 3: Schematic for the time evolution scheme, in $1+1$ dimensions, for a two-grid hierarchy. The large filled (red) circles represent data on the coarse grid, and smaller filled (green) circles represent data on the fine grid. The algorithm uses the following order. 1: Coarse grid time step, 2 and 3: fine grid time steps, 4: restriction from fine grid to coarse grid. Since the fine grid is always nested inside a coarse grid, there are also coarse grid points (not shown) spanning the fine grid region (at times when the coarse grid is defined) at the locations of "every other" fine grid point; the data at these coarse grid points are restricted (copied directly) from the fine grid data.
• Figure 4: Schematic for the "buffering" during time integration. Shown is the left edge of a refined region, which extends further to the right, which is integrated in time with a 3-step ICN method. At the filled points (in the interior), time integration proceeds as usual. The empty points (near the boundary) are left out, because no boundary condition is given during time integration. A prolongation after the time integration fills the empty points again. This whole scheme corresponds to either of the steps labelled 2 and 3 in Figure \ref{['fig:Timestepping']}.
• Figure 5: Schematic for initial data scheme, in $1+1$ dimensions. Our use of quadratic interpolation in time requires three time levels of coarse grid data in order to provide boundary data for evolution on fine grids. To achieve this from the beginning of the evolution (without the use of a known continuum solution with which to "pre-load" these levels), we evolve our initial data (defined at $t=0$) both forwards and backwards one step in time. In this way, three time levels of coarse grid data are always available to provide boundary data along the edges of fine grids. The data at various times are denoted by fractions of the time step $\Delta t^0$ on the base grid. The coarsest grid is shown as by a solid (red) line, a finer grid by a long-dashed (green) line, and a still finer grid by a dotted (blue) line. (We perform some additional backwards evolution as well, which we describe in the main text. The essence of the scheme, however, is given here.)