Time-symmetric ADI and causal reconnection

Summary

Abstract. We study the effect of a moving grid on the stability of the finite difference approximations to the wave equation. We introduce two techniques, which we call “causal reconnection” and “time-symmetric ADI” that together provide efficient, accurate and stable integration schemes for all grid velocities in any number of dimensions.

INTRODUCTION

In the numerical study of wave phenomena it is often necessary to use a reference frame that is moving with respect to the medium in which the waves propagate. In this paper, by studying the simple wave equation, we show that the consistent application to such a problem of two fundamental physical principles — causality and time-reversal-invariance — produces remarkably stable, efficient and accurate integration methods.

Our principal motivation for studying these techniques is the development of algorithms for the numerical simulation of moving, interacting black-holes. If we imagine a black hole moving “through” a finite difference grid then some requirements become clear. As the hole moves, grid points ahead of it will fall inside the horizon, while others will emerge on the other side. This requires grids that shift faster than light. Moreover, in situations when the dynamical time scale is large, one would like to be free of the Courant stability condition on time-steps, i.e. one wants to use implicit methods. Full implicit schemes require the inversion of huge sparse matrices. Alternating Direction Implicit (ADI) schemes reduce the computational burden by turning the integration into a succession of one-dimensional implicit integrations.