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Geometric integration of Hamiltonian systems perturbed by Rayleigh damping

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Abstract

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter ε, and the schemes under study preserve the symplectic structure in the case ε=0. In the case 0<ε≪1 the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted.

Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order.

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Authors and Affiliations

  1. School of Engineering and Advanced Technology, Massey University, Private Bag 11 222, Palmerston North, 4442, New Zealand

    Klas Modin

  2. Centre for Mathematical Sciences, Lund University, Box 118, 221 00, Lund, Sweden

    Gustaf Söderlind

Authors
  1. Klas Modin
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  2. Gustaf Söderlind
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Correspondence to Klas Modin.

Additional information

Communicated by Christian Lubich.

Presented at the BIT50 conference in Lund, Sweden, 17–20 June 2010.

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Modin, K., Söderlind, G. Geometric integration of Hamiltonian systems perturbed by Rayleigh damping. Bit Numer Math 51, 977–1007 (2011). https://doi.org/10.1007/s10543-011-0345-1

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  • DOI: https://doi.org/10.1007/s10543-011-0345-1

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