Journal of Scientific Computing

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

Important aspects of geometric numerical integration

  • Ernst HairerEmail author


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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