THE CONTROL OF PARASITISM IN G-SYMPLECTIC METHODS

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...

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Published inSIAM journal on numerical analysis Vol. 52; no. 5; pp. 2440 - 2465
Main Authors BUTCHER, JOHN C., HABIB, YOUSAF, HILL, ADRIAN T., NORTON, TERENCE J. T.
Format Journal Article
LanguageEnglish
Published Society for Industrial and Applied Mathematics 01.01.2014
Subjects
Angular momentum
Computer simulation
Design engineering
Differential equations
Invariants
Mathematical models
Mathematical sequences
Numerical analysis
Numerical integration
Ordinary differential equations
Parasitism
Pendulums
Perturbation methods
Preserves
Remedies
Rigid structures
Runge Kutta method
Tableaux
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Summary: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.
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ISSN:0036-1429
1095-7170
DOI:10.1137/140953277