Black Hole Excision with Multiple Grid Patches

When using black hole excision to numerically evolve a black hole spacetime with no continuous symmetries, most 3+1 finite differencing codes use a Cartesian grid. It’s difficult to do excision on such a grid because the natural r = constant excision surface must be approximated either by a very different shape such as a contained cube, or by an irregular and non-smooth “LEGO sphere” which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme, which uses multiple {r × (angular coordinates)} grid patches, each patch using a different (nonsingular) choice of angular coordinates. This allows excision on a smooth r = constant 2-sphere. I discuss the key design choices in such a multiple-patch scheme, including the choice of ghost-zone versus internal-boundary treatment of the interpatch boundaries (I use a ghost-zone scheme), the number and shape of the patches (I use a 6-patch “inflated-cube” scheme), the details of how the ghost zones are “synchronized” by interpolation from neighboring patches, the tensor basis for the Einstein equations in each patch, and the handling of non-tensor field variables such as the BSSN Γ̃ (I use a scheme which requires ghost zones which are twice as wide for the BSSN conformal factor φ as for Γ̃ and the other BSSN field variables). I present sample numerical results from a prototype implementation of this scheme. This code simulates the time evolution of the (asymptotically flat) spacetime around a single (excised) black hole, using 4th order finite differencing in space and time. Using Kerr initial data with J/m = 0.6, I present evolutions to t ∼ 1500m. The lifetime of these evolutions appears to be limited only by outer boundary instabilities, not by any excision instabilities or by any problems inherent to the multiple-patch scheme.