  It is expected that the realization of a convergent and long-term stable
numerical code for the simulation of a black hole inspiral collision will
depend greatly upon the construction of stable algorithms capable of handling
smooth and, most likely, time dependent boundaries. After deriving single grid,
energy conserving discretizations for axisymmetric systems containing the axis
of symmetry, we present a new excision method for moving black holes using
multiple overlapping coordinate patches, such that each boundary is fixed with
respect to at least one coordinate system. This multiple coordinate structure
eliminates all need for extrapolation, a commonly used procedure for moving
boundaries in numerical relativity.
  We demonstrate this excision method by evolving a massless Klein-Gordon
scalar field around a boosted Schwarzschild black hole in axisymmetry. The
excision boundary is defined by a spherical coordinate system co-moving with
the black hole. Our numerical experiments indicate that arbitrarily high boost
velocities can be used without observing any sign of instability.
