On the qualitative behaviour of symplectic integrators. II: Integrable systems
type: Item
instance of: scholarly article
title: On the qualitative behaviour of symplectic integrators. II: Integrable systems
zbMATH Open document ID: 0892.65040
author: Daniel Stoffer
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-03-24T00:00:00Z
review text: [For part I see Numer. Math. 77, No. 4, 535-547 (1997, Zbl 0885.65088).] This paper deals with the qualitative error behaviour of symplectic one step integrators on completely integrable Hamiltonian systems. As it is well konwn, \textit{N. N. Nekhoroshev} [Russ. Math. Surveys, 32, No. 6, 1-65 (1977; Zbl 0389.70028)] derived stability bounds which hold on exponentially long time intervals for perturbed Hamiltonian systems. Since the solution of a symplectic method can be considered as the exact solution of a perturbed Hamiltonian it is conceivable that under suitable assumptions some Nekhoroshev type results could be established for symplectic integrators. In this paper, the author proves that for completely integrable Hamiltonian systems all first integrals are approximately preserved (in the sense that the errors grow linearly) for exponentially long times. This implies that the global error of symplectic integrators grows linearly for such intervals in contrast with nonsymplectic integrators in which global errors grow in general quadratically.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L70
zbMATH DE Number: 1115483
zbMATH Keywords: first integrals preservation
zbMATH Keywords: error bounds
zbMATH Keywords: completely integrable Hamiltonian systems
zbMATH Keywords: stability bounds
zbMATH Keywords: symplectic method
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5728
OpenAlex ID: W2059479268
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DOI: 10.1006/JMAA.1997.5728
description: scientific article; zbMATH DE number 1115483