Abstract
Conventional variable-step implementation of symplectic or reversible integration methods destroy the symplectic or reversible structure of the system. We show that to preserve the symplectic structure of a method the step size has to be kept almost constant. For reversible methods variable steps are possible but the step size has to be equal for “reflected” steps. We demonstrate possible ways to construct reversible variable step size methods. Numerical experiments show that for the Kepler problem the new methods perform better than conventional variable step size methods or symplectic constant step size methods. In particular they exhibit linear growth of the global error (as symplectic methods with constant step size).
Zusammenfassung
Werden symplektische oder reversible Integrationsverfahren mit herkömmlichen Schrittweitensteuerungen verwendet, so geht die symplektische, bzw. die reversible Struktur des Problems verloren. In dieser Arbeit wird gezeigt, dass die symplektische Struktur des Verfahrens nur dann erhalten wird, wenn die Schrittweite fast konstant bleibt. Für reversible Verfahren sind echt variable Schrittweiten möglich, die Schrittweite muss jedoch für “gespiegelte” Schritte gleich sein. Es werden verschiedene Wege aufgezeigt, um reversible, variable Schrittweiten zu konstruieren. Numerische Experimente zeigen, dass für das Keplerproblem die neuen Methoden den herkömmlichen Schrittweitensteuerungen oder den symplektischen Verfahren mit konstanter Schrittweite überlegen sind. Insbesondere wächst der globale Fehler linear.
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Stoffer, D. Variable steps for reversible integration methods. Computing 55, 1–22 (1995). https://doi.org/10.1007/BF02238234
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DOI: https://doi.org/10.1007/BF02238234