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Lokale Fehlerschätzung mittels modifizierter Richardson-Extrapolation in linear impliziten Einschrittverfahren

Estimation of the local error by a modified Richardson extrapolation in linear Implicit One-Step Methods

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Zusammenfassung

Für die Schätzung des lokalen Diskretisierungsfehlers in Rosenbrock-Wanner-und adaptiven Runge-Kutta-Methoden wird eine modifizierte Richardson-Extrapolation angegeben, die eine wesentliche Aufwandsenkung pro Schritt ergibt. Die Grundidee dieser Methode findet man bei Bui/Poon [1]. In der dort angegebenen Form ist diese Methode nur für lineare Differentialgleichungssysteme korrekt. Ziel unserer Arbeit ist eine korrekte Version für nichtlineare steife Systeme anzugeben. Beispiele stark Astabiler ROW-Methoden und ARK-Verfahren mit modifizierter Richardson-Extrapolation werden mitgeteilt.

Abstract

In this paper we describe a modified Richardson extrapolation for error estimate in a Rosenbrock-Wanner procedure and in an adaptive Runge-Kutta procedure. This method requires only very little extra computation to calculate the solution with step-length 2h. The basic idea of this method is described in Bui/Poon [1]. However the method of Bui/Poon is only correct for linear differential equations. The aim of our paper is to present a correct version for nonlinear stiff systems. Examples of strong A-stable ROW-and ARK-procedures with modified Richardson extrapolation are presented.

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Literatur

  1. Bui, T. D., Poon, S. W. H.: On the computational aspects of Rosenbrock procedures with built-in error estimates for stiff systems. BIT21, 168–174 (1981).

    Article  Google Scholar 

  2. Cash, J. R.: Semi-implicit Runge-Kutta procedures with error estimates for the numerical integration of stiff systems of ordinary differential equations. J. ACM23, 455–460 (1976).

    Article  Google Scholar 

  3. Kaps, P.: Rosenbrock-type methods. In: Numerical methods for solving stiff initial value problems (Dahlquist, G., Jeltsch, R., eds.). [Bericht Nr. 9 (1981) RWTH Aachen, Inst. für Geometrie und Prakt. Math.]

  4. Kaps, P., Wanner, G.: A study of Rosenbrock-type methods of high order. Numer. Math.38, 279–298 (1981).

    Article  Google Scholar 

  5. Kaps, P., Rentrop, R.: Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. Numer. Math.33, 55–68 (1979).

    Article  Google Scholar 

  6. Strehmel, K., Weiner, R.: Behandlung steifer Anfangswertprobleme gewöhnlicher Differentialgleichungen mit adaptiven Runge-Kutta-Methoden. Computing29, 153–165 (1982).

    Google Scholar 

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Authors and Affiliations

  1. Sektion Mathematik, Martin-Luther-Universität Halle-Wittenberg, Weinbergweg 17, DDR-4020, Halle a. d. Saale, Deutsche Demokratische Republik

    K. Strehmel &  R. Weiner

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  1. K. Strehmel
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  2. R. Weiner
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Strehmel, K., Weiner, R. Lokale Fehlerschätzung mittels modifizierter Richardson-Extrapolation in linear impliziten Einschrittverfahren. Computing 33, 131–140 (1984). https://doi.org/10.1007/BF02240184

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  • DOI: https://doi.org/10.1007/BF02240184

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