Reconsidering trigonometric integrators

Robert I McLachlan, Dion R O'Neale

Abstract


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 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.

doi:10.1017/S1446181109000042

Keywords


numerical integration, initial value problems, highly oscillatory Hamiltonian systems, trigonometric integrators, symplectic algorithms.



DOI: http://dx.doi.org/10.21914/anziamj.v50i0.2318



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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.