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Journal of Scientific Computing

, Volume 25, Issue 1–2, pp 67–81 | Cite as

Important aspects of geometric numerical integration

  • Ernst HairerEmail author
Article

Abstract

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.

Key words

Geometric numerical integration Hamiltonian systems reversible differential equations backward error analysis energy conservation modulated Fourier expansion adiabatic invariants sine-Gordon equation 

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Copyright information

© Springer Science+Business Media, Inc 2005

Authors and Affiliations

  1. 1.Section de MathématiquesGenève 24Switzerland

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