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RECONSIDERING TRIGONOMETRIC INTEGRATORS

Part of: Numerical problems in dynamical systems Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  03 November 2009

DION R. J. O’NEALE  and
ROBERT I. MCLACHLAN
DION R. J. O’NEALE*
Affiliation:
Institute of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North, New Zealand (email: [email protected], [email protected])
ROBERT I. MCLACHLAN
Affiliation:
Institute of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North, New Zealand (email: [email protected], [email protected])
*
✉For correspondence; e-mail: [email protected], [email protected]
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Abstract

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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.

Keywords

numerical integrationinitial value problemshighly oscillatory Hamiltonian systemstrigonometric integratorssymplectic algorithms

MSC classification

Secondary: 65L05: Initial value problems 65P40: Nonlinear stabilities
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 3: This Special Issue is dedicated to Dr Stephen White , January 2009 , pp. 320 - 332
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Arnold, V. I., Mathematical methods of classical mechanics, 2nd edn, Volume 60 of Graduate Texts in Mathematics (Springer-Verlag, Berlin, 1989).Google Scholar
[2]Le Bris, C. and Legoll, F., “Dérivation de schémas numériques symplectiques pour des systèmes hamiltoniens hautement oscillants”, C. R. Acad. Sci. Paris, Ser I 344 (2007) 277–282.CrossRefGoogle Scholar
[3]Dueflhard, P., “A study of extrapolation methods based on multistep schemes without parasitic solutions”, Z. Angew. Math. Phys. 30 (1979) 177–189.CrossRefGoogle Scholar
[4]García-Archilla, B., Sanz-Serna, J. M. and Skeel, R. D., “Long-time-step methods for oscillatory differential equations”, SIAM J. Sci. Comput. 20 (1998) 930–963.CrossRefGoogle Scholar
[5]Gautschi, W., “Numerical integration of ordinary differential equations based on trigonometric polynomials”, Numer. Math. 3 (1961) 381–397.CrossRefGoogle Scholar
[6]Grimm, V. and Hochbruck, M., “Error analysis of exponential integrators for oscillatory second-order differential equations”, J. Phys. A: Math. Gen. 39 (2006) 5495–5507.CrossRefGoogle Scholar
[7]Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields (Springer-Verlag, New York, 1983).CrossRefGoogle Scholar
[8]Hairer, E. and Lubich, Ch., “Long-time energy conservation of numerical methods for oscillatory differential equations”, SIAM J. Numer. Anal. 38 (2000) 414–441.CrossRefGoogle Scholar
[9]Hairer, E., Lubich, Ch. and Wanner, G., Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, 2nd edn (Springer-Verlag, Berlin, 2002).CrossRefGoogle Scholar
[10]Hochbruck, M. and Lubich, Ch., “A Gautschi-type method for oscillatory second-order differential equations”, Numer. Math. 83 (1999) 403–426.CrossRefGoogle Scholar
[11]Sauer, T. and York, J. A., “Rigorous verification of trajectories for the computer simulation of dynamical systems”, Nonlinearity 4 (1994) 961–979.CrossRefGoogle Scholar
[12]Skeel, R. D. and Srinivas, K., “Nonlinear stability analysis of area-preserving integrators”, SIAM J. Numer. Anal. 38 (2000) 129–148.CrossRefGoogle Scholar