Stable 3-level leapfrog integration in numerical relativity

The 3-level leapfrog time integration algorithm is an attractive choice for numerical relativity simulations since it is time symmetric and avoids non-physical damping. In Newtonian problems without velocity dependent forces, this method enjoys the advantage of long term stability. However, for more general differential equations, whether ordinary or partial, delayed onset numerical instabilities can arise and destroy the solution. A known cure for such instabilities appears to have been overlooked in many application areas. We give an improved cure (``deloused leapfrog'') that both reduces memory demands important for (3+1) -dimensional wave equations and allows for the use of adaptive time steps without a loss in accuracy. We show both that the instability arises and that the cure we propose works in highly relativistic problems such as tightly bound geodesics, spatially homogeneous spacetimes, and strong gravitational waves. In the gravitational wave test case (polarized waves in a Gowdy spacetime) the deloused leapfrog method was five to eight times less CPU costly at various accuracies than the implicit Crank-Nicholson method, which is not subject to this instability.