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It is expected that the realization of a convergent and long-term stablenumerical code for the simulation of a black hole inspiral collision willdepend greatly upon the construction of stable algorithms capable of handlingsmooth and, most likely, time dependent boundaries. After deriving single grid,energy conserving discretizations for axisymmetric systems containing the axisof symmetry, we present a new excision method for moving black holes usingmultiple overlapping coordinate patches, such that each boundary is fixed withrespect to at least one coordinate system. This multiple coordinate structureeliminates all need for extrapolation, a commonly used procedure for movingboundaries in numerical relativity.  We demonstrate this excision method by evolving a massless Klein-Gordonscalar field around a boosted Schwarzschild black hole in axisymmetry. Theexcision boundary is defined by a spherical coordinate system co-moving withthe black hole. Our numerical experiments indicate that arbitrarily high boostvelocities can be used without observing any sign of instability.Comment: 23 pages, 13 figures. Revised versio

Spherical excision for moving black holes and summation by parts for axisymmetric systems