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Long-term analysis of the Störmer–Verlet method for Hamiltonian systems with a solution-dependent high frequency

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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.

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Authors and Affiliations

  1. Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211, Geneva 4, Switzerland

    Ernst Hairer

  2. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle, 72076, Tübingen, Germany

    Christian Lubich

Authors
  1. Ernst Hairer
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  2. Christian Lubich
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Correspondence to Christian Lubich.

Additional information

This work has been supported by the Fonds National Suisse, Project No. 200020-144313/1.

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Hairer, E., Lubich, C. Long-term analysis of the Störmer–Verlet method for Hamiltonian systems with a solution-dependent high frequency. Numer. Math. 134, 119–138 (2016). https://doi.org/10.1007/s00211-015-0766-x

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  • DOI: https://doi.org/10.1007/s00211-015-0766-x

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