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 semidiscrete scheme if the shift parameter is too large. This implies that discretizations obtained using integrators such as Runge-Kutta, Crank-Nicholson, and leapfrog 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 Arnowitt-Deser-Misner-type variables is very effective.

• Received 14 October 2004

DOI:https://doi.org/10.1103/PhysRevD.71.027501