Authors

Ernst Hairer, Christian Lubich

Abstract

The long-time behaviour of the Störmer–Verlet–leapfrog method is studied when this method is applied to highly oscillatory Hamiltonian systems with a slowly varying, solution-dependent high frequency. Using the technique of modulated Fourier expansions with state-dependent frequencies, which is newly developed here, the following results are proved: the considered Hamiltonian systems have the action as an adiabatic invariant over long times that cover arbitrary negative powers of the small parameter. The Störmer–Verlet method approximately conserves a modified action and a modified total energy over a long time interval that covers a negative integer power of the small parameter. This power depends on the size of the product of the stepsize with the high frequency.

Keywords

65P10; 65L05; 34E13

Citation

  • Journal: Numerische Mathematik
  • Year: 2016
  • Volume: 134
  • Issue: 1
  • Pages: 119–138
  • Publisher: Springer Science and Business Media LLC
  • DOI: 10.1007/s00211-015-0766-x

BibTeX

@article{Hairer_2015,
  title={{Long-term analysis of the Störmer–Verlet method for Hamiltonian systems with a solution-dependent high frequency}},
  volume={134},
  ISSN={0945-3245},
  DOI={10.1007/s00211-015-0766-x},
  number={1},
  journal={Numerische Mathematik},
  publisher={Springer Science and Business Media LLC},
  author={Hairer, Ernst and Lubich, Christian},
  year={2015},
  pages={119--138}
}

Download the bib file

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