RECONSIDERING TRIGONOMETRIC INTEGRATORS

DION R. J. O'NEALE, ROBERT I. MCLACHLAN
2009 ANZIAM journal (Print)  
In this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi-Pasta-Ulam problem, a highly oscillatory Hamiltonian system. We also show
more » ... stem. We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but at the wrong values. We use statistical properties such as time averages to further evaluate the performance of the trigonometric methods and compare the performance with that of the mid-point rule. 2000 Mathematics subject classification: primary 65L05; secondary 65P10, 65P40.
doi:10.1017/s1446181109000042 fatcat:37kbx25qg5fzdmmhwv6gt4sk3u