Authors

Ernst Hairer, Christian Lubich, Gerhard Wanner

Abstract

The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved long-time behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a cross-section of the recent monograph by the authors, enriched by some additional material.After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent long-time behaviour of the method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, and preservation of adiabatic invariants.

Citation

  • Journal: Acta Numerica
  • Year: 2003
  • Volume: 12
  • Issue:
  • Pages: 399–450
  • Publisher: Cambridge University Press (CUP)
  • DOI: 10.1017/s0962492902000144

BibTeX

@article{Hairer_2003,
  title={{Geometric numerical integration illustrated by the Störmer–Verlet method}},
  volume={12},
  ISSN={1474-0508},
  DOI={10.1017/s0962492902000144},
  journal={Acta Numerica},
  publisher={Cambridge University Press (CUP)},
  author={Hairer, Ernst and Lubich, Christian and Wanner, Gerhard},
  year={2003},
  pages={399--450}
}

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