Current methods of evolving a spacetime containing one or more black holes
are plagued by instabilities that prohibit long-term evolution. Some of the
se instabilities may be due to the numerical method used, traditionally fin
ite differencing. In this paper, we explore the use of a pseudospectral col
location (PSC) method for the evolution of a spherically symmetric black ho
le spacetime in one dimension using a hyperbolic formulation of Einstein's
equations. We demonstrate that our PSC method is able to evolve a spherical
ly symmetric black hole spacetime forever without enforcing constraints, ev
en if we add dynamics via a Klein-Gordon scalar field. We find that, in con
trast with finite-differencing methods, black hole excision is a trivial op
eration using PSC applied to a hyperbolic formulation of Einstein's equatio
ns. We discuss the extension of this method to three spatial dimensions.