Long-Time-Step Methods for Oscillatory Differential Equations

Considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present. It is assumed that one can do an explicit separation of the RHS "forces" into fast forces and slow forces such that (i) the fast forces contain the high frequency part of the solution, (ii) the fast forces are conservative, and (iii) the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem. The fast forces are allowed to have low frequency components. Particular applications for which the schemes are intended include N-body problems (for which most of the forces are slow) and nonlinear wave phenomena (for which the fast forces can be propagated by spectral methods). The assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces. A long-time-step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system. An existing method that comes close to qualifying is the "impulse" method, also known as Verlet-I and r-RESPA. It is shown that it might fail, though, for a couple of reasons. First, it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode, and, second, it is unstable if the step size is near a multiple of half the period of a normal mode. Proposed in this paper is a "mollified" impulse method having an error bound that is independent of the frequency of the fast forces. It is also shown to possess superior stability properties. Theoretical results are supplemented by numerical experiments. The method is efficient and reasonably easy to implement.