Scientific article
Open access

Important aspects of geometric numerical integration

ContributorsHairer, Ernst
Published inJournal of scientific computing, vol. 25, no. 1/2, p. 67-81
Publication date2005

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.

  • Geometric numerical integration
  • Hamiltonian systems
  • Reversible differential equations
  • Backward error analysis
  • Energy conservation
  • Modulated Fourier expansion
  • Adiabatic invariants
  • Sine-Gordon equation
Citation (ISO format)
HAIRER, Ernst. Important aspects of geometric numerical integration. In: Journal of scientific computing, 2005, vol. 25, n° 1/2, p. 67–81. doi: 10.1007/s10915-004-4633-7
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Article (Published version)
ISSN of the journal0885-7474

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