The control of parasitism in $G$-symplectic methods

John C. Butcher, Yousaf Habib, Adrian T. Hill, Terence J T Norton

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26 Citations (SciVal)
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G-symplectic general linear methods are designed to approximately preserve symplectic invariants for Hamiltonian systems. In this paper, the properties of G-symplectic methods are explored computationally and theoretically. Good preservation properties are observed over long times for many parameter ranges, but, for other parameter values, the parasitic behavior, to which multivalue methods are prone, corrupts the numerical solution by the growth of small perturbations. Two approaches for alleviating this effect are considered. First, compositions of methods with growth parameters of opposite signs can be used to cancel the long-term effect of parasitism. Second, methods can be constructed for which the growth parameters are zero by design. Each of these remedies is found to be successful in eliminating parasitic behavior in long-term simulations using a variety of test problems.

Original languageEnglish
Pages (from-to)2440-2465
Number of pages26
JournalSIAM Journal on Numerical Analysis (SINUM)
Issue number5
Publication statusPublished - 14 Oct 2014


  • General linear methods
  • Parasitism
  • Symplectic


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