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Important Aspects of Geometric Numerical Integration

Hairer, Ernst

In: Journal of Scientific Computing, 2005, vol. 25, no. 1, p. 67-81

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    Title
    • Important Aspects of Geometric Numerical Integration
    Author
    • Hairer, Ernst. Section de Mathématiques, 2-4, rue du Lièvre, CH-1211, Genève 24, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Journal of Scientific Computing, 2005, vol. 25, no. 1, p. 67-81. Kluwer Academic Publishers-Plenum Publishers
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:321742
    Summary
    At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods). In the ‘non-stiff' situation the long-time behaviour of these methods is well-understood and can be explained with the help of a backward error analysis. In the highly oscillatory (‘stiff') case this theory breaks down. Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high frequencies are present. This paper terminates with numerical experiments at space discretizations of the sine-Gordon equation, where a whole spectrum of frequencies is present