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Important aspects of geometric numerical integration

Published in Journal of Scientific Computing. 2005, vol. 25, no. 1/2, p. 67-81
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.
Keywords Geometric numerical integrationHamiltonian systemsReversible differential equationsBackward error analysisEnergy conservationModulated Fourier expansionAdiabatic invariantsSine-Gordon equation.
Stable URL http://archive-ouverte.unige.ch/unige:12118
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Deposited on : 2010-10-15

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