RECONSIDERING TRIGONOMETRIC INTEGRATORS

• Published in: The ANZIAM journal Vol. 50; no. 3; pp. 320 - 332
• Main Authors: O’NEALE, DION R. J., MCLACHLAN, ROBERT I.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.01.2009
• Subjects: 65P40; Differential equations; Hamiltonian functions; highly oscillatory Hamiltonian systems; initial value problems; Integrators; numerical integration; Performance evaluation; primary 65L05; secondary 65P10; symplectic algorithms; trigonometric integrators; primary 65L05; secondary 65P10; numerical integration; initial value problems; highly oscillatory Hamiltonian systems; symplectic algorithms; trigonometric integrators; 65P40
• ISSN: 1446-1811; 1446-8735
• DOI: 10.1017/S1446181109000042

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.