  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 $r = \text{constant}$ excision surface must be approximated either by a
very different shape such as a contained cube, or by an irregular and
non-smooth "LEGO(tm) sphere" which may introduce numerical instabilities into
the evolution. In this paper I describe an alternate scheme, which uses
multiple $\{r \times (\text{angular coordinates}) \}$ grid patches, each patch
using a different (nonsingular) choice of angular coordinates. This allows
excision on a smooth $r = \text{constant}$ 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, the number and shape of the patches, 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 $\tilde{\Gamma}^i$.
  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 $J/m^2 = 0.6$, I
present evolutions to $t \gtsim 1500m$. 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.
