Time-Symmetric ADI and Causal Reconnection: Stable Numerical Techniques for Hyperbolic Systems on Moving Grids

Abstract

Moving grids are of interest in the numerical solution of hydrodynamical problems and in numerical relativity. We show that conventional integration methods for the simple wave equation in one and more than one dimension exhibit a number of instabilities on moving grids. We introduce two techniques, which we call causal reconnection and time-symmetric ADI, which together allow integration of the wave equation with absolute local stability in any number of dimensions on grids that may move very much faster than the wave speed and that can even accelerate. These methods allow very long time-steps, are fully second-order accurate, and offer the computational efficiency of operator-splitting. We develop causal reconnection first in the one-dimensional case; we find that a conventional implicit integration scheme that is unconditionally stable as long as the speed of the grid is smaller than that of the waves nevertheless turns unstable whenever the grid speed increases beyond this value. We introduce a notion of local stability for difference equations with variable coefficients. We show that, by "reconnecting" the computational molecule at each time-step in such a way as to ensure that its members at different time-steps are within one another's causal domains, one eliminates the instability, even if the grid accelerates. This permits very long time-steps on rapidly moving grids. The method extends in a straightforward and efficient way to more than one dimension. However, in more than one dimension, it is very desirable to use operator-splitting techniques to reduce the computational demands of implicit methods, and we find that standard schemes for integrating the wave equation—Lees' first and second alternating direction implicit (ADI) methods—go unstable for quite small grid velocities. Lees' first method, which is only first-order accurate on a shifting grid, has mild but nevertheless significant instabilities. Lees' second method, which is second-order accurate, is very unstable. By adopting a systematic approach to the design of ADI schemes, we develop a new ADI method that cures the instability for all velocities in any direction up to the wave speed. This scheme is uniquely defined by a simple physical principle: the ADI difference equations should be invariant under time-inversion. (The wave equation itself and the fully implicit difference equations satisfy this criterion, but neither of Lees' methods do.) This new time-symmetric ADI scheme is, as a bonus, second-order accurate. It is thus far more efficient than a fully implicit scheme, just as stable, and just as accurate. By implementing causal reconnection of the computational molecules, we extend the time-symmetric ADI scheme to arrive at a scheme that is second-order accurate, computationally efficient, and unconditionally locally stable for all grid speeds and long time-steps. We have tested the method by integrating the wave equation on a rotating grid, where it remains stable even when the grid speed at the edge is 15 times the wave speed. Because our methods are based on simple physical principles, they should generalize in a straightforward way to many other hyperbolic systems. We discuss briefly their application to general relativity and their potential generalization to fluid dynamics.