Abstract
Runge-Kutta-Nyström type methods and special predictor-corrector methods are constructed for the accurate solution of second-order differential equations of which the solution is dominated by the forced oscillation originating from an external, periodic forcing term. For a family of second-order explicit and linearly implicit Runge-Kutta-Nyström methods it is shown that the forced oscillation is represented with zero phase lag. For a family of predictor-corrector methods of fourth-order, it is shown that both the phase lag order and the dissipation of the forced oscillation can be made arbitrarily high. Numerical examples illustrate the effectiveness of our reduced phase lag methods.
Zusammenfassung
Für die numerische Behandlung von Differentialgleichungen zweiter Ordnung, bei dennen die Lösung im wesentlichen durch die von einer äußeren periodischen Kraft erzwungenen Schwingung bestimmt wird, werden Diskretisierungsmethoden vom Runge-Kutta-Nyström Typ und spezielle Prädiktor-Korrektor Methoden konstruiert. Für eine Klasse expliziter und linear-impliziter Runge-Kutta-Nyström Methoden der Ordung zwei zeigen wir, daß die erzwungene Schwingung keinen Phasenfehler aufweist. Für eine Klasse von Prädiktor-Korrektor Methoden vierter Ordnung wird nachgewiesen, daß die Phasen- und Dissipationsfehlerordnung beliebig groß gemacht werden kann. Numerische Beispiele bestätigen die Wirksamkeit unserer Methoden mit reduziertem Phasenfehler.
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van der Houwen, P.J., Sommeijer, B.P., Strehmel, K. et al. On the numerical integration of second-order initial value problems with a periodic forcing function. Computing 37, 195–218 (1986). https://doi.org/10.1007/BF02252512
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DOI: https://doi.org/10.1007/BF02252512