  Keeping Einstein's equations in second order form can be appealing for
computational efficiency, because of the reduced number of variables and
constraints. Stability issues emerge, however, which are not present in first
order formulations. We show that a standard discretization of the second order
``shifted'' wave equation leads to an unstable semi-discrete scheme if the
shift parameter is too large. This implies that discretizations obtained using
integrators such as Runge-Kutta, Crank-Nicholson, leap-frog are unstable for
any fixed value of the Courant factor. We argue that this situation arises in
numerical relativity, particularly in simulations of spacetimes containing
black holes, and discuss several ways of circumventing this problem. We find
that the first order reduction in time based on ``ADM'' type variables is very
effective.

General Relativity and Quantum Cosmology