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 is 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 'LEGO1 sphere' which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme which uses multiple {r x (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 neighbouring patches, the tensor basis for the Einstein equations in each patch, and the handling of non-tensor field variables such as the BSSN Γ-tildei (I use a scheme which requires ghost zones which are twice as wide for the BSSN conformal factor φ as for Γ-tildei 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 fourth-order finite differencing in space and time. Using Kerr initial data with J/m2 = 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