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Consistency and convergence of general linear multistep variable stepsize variable formula methods

Konsistenz und Konvergenz allgemeiner linearer Mehrschrittverfahren mit variabler Schrittweite und variabler Formel

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Abstract

Linear multistep (LM) formulae are commonly used in the numerical solution of initial value problems of first order ordinary differential equations (ODE's). A rigorous theory for LM formulae, when these are implemented as constant stepsize constant formula methods, was developed after the publication of Dahlquist's classical paper [1] in 1956. After 1969 LM formulae have often been applied in practical codes as variable stepsize variable formula methods (VSVFM's). Therefore the development of a rigorous theory for LM formulae also in the case where these are used as VSVFM's is desirable. A formal definition of general LM VSVFM's is given in this paper. Then some theorems concerning the consistency and the convergence of general LM VSVFM's are formulated and proved. The results obtained in this paper can be extended for one-leg VSVFM's and for VSVFM's based on predictorcorrector schemes of different types.

Zusammenfassung

Lineare Mehrschrittverfahren werden zur numerischen Lösung von Anfangswertproblemen für Systeme erster Ordnung gewöhnlicher Differentialgleichungen angewandt. Eine strenge Theorie für lineare Mehrschrittverfahren, wenn diese mit konstanter Schrittweite und konstanter Formel implementiert werden, wurde nach der Publikation des klassischen Artikels von G. Dahlquist ([1]) entwickelt. Seit 1969 werden lineare Mehrschrittverfahren häufig in Codes mit variabler Schrittweite und variabler Formel verwendet. Deswegen ist eine Entwicklung einer strengen Theorie für lineare Mehrschrittverfahren mit variabler Schrittweite und variabler Formel wünschenswert. Eine formale Definition allgemeiner linearer Mehrschrittverfahren mit variabler Schrittweite und variabler Formel wird in diesem Artikel gegeben. Dann werden einige Theoreme über die Konsistenz und Konvergenz allgemeiner linearer Mehrschrittverfahren mit variabler Schrittweite und variaber Formel formuliert und bewiesen. Die Resultate in diesem Artikel können auf “one-leg”-Methoden und auf Prädiktor-Korrektor-Methoden verschiedener Typen mit variabler Schrittweite und variabler Formel ausgedehnt werden.

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  1. Risø National Laboratory, Air Pollution Laboratory National Agency of Environmental Protection, DK-4000, Roskilde, Denmark

    Z. Zlatev

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Zlatev, Z. Consistency and convergence of general linear multistep variable stepsize variable formula methods. Computing 31, 47–67 (1983). https://doi.org/10.1007/BF02247936

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