A Multiple-Grid-Patch Evolution Scheme for 3-D Black Hole Excision

When using black hole excision to numerically evolve a fully generic black hole spacetime, most 3-D 3+1 codes use an $xyz$-topology (spatial) grid. In such a grid, an $r=\text{constant}$ excision surface must be approximated by an irregular and non-smooth “staircase-shaped” excision grid boundary, which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme, which uses multiple grid patches, each with topology ${r \times ({\rm angular coordinates})}$,tocoverthesliceoutsidethe$r = \constant$ excision surface. The excision grid boundary is now smooth, so the evolution should be less prone to instabilities. With 4th order finite differencing, this code evolves Kerr initial data to $\sim 60M$ using the ADM equations; I’m currently implementing the BSSN equations in it in the hope that this will improve the stability.