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Important aspects of geometric numerical integration

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Ernst Hairer

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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.

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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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