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, Volume 34, Issue 1, pp 17–40 | Cite as

Rosenbrock methods for Stiff ODEs: A comparison of Richardson extrapolation and embedding technique

  • P. Kaps
  • S. W. H. Poon
  • T. D. Bui
Article

Abstract

In [16], Rosenbrock methods of order four are investigated using an embedded method of order three for step size control. Here, we study such a method using Richardson extrapolation for step size control and compare the two techniques with each other. The usual belief that extrapolation is inferior to embedding is not true. Numerical results for the 25 examples of STIFF DETEST and for some more difficult problems show the following behaviour: For low tolerances (∼10−2) embedding is superior, for moderate tolerances (∼10−4) both techniques are comparable and for high tolerances (<10−5) extrapolation is superior. Under certain conditions the extrapolated value can be used for step continuation without stability problems.

AMS Subject Classification

65L05 

Key words

Rosenbrock methods semi-implicit Runge-Kutta methods Richardson extrapolation embedding technique 

Ein Vergeich von Richardsonextrapolation und Einbettungstechnik für Rosenbrockmethoden

Zusammenfassung

In [16] werden Rosenbrockmethoden der Ordnung 4 untersucht, die zur Schrittweitensteuerung eine eingebettete Methode der Ordnung 3 verwenden. In dieser Arbeit werden neue solche Methoden hergeleitet, die auf Schrittweitensteuerung durch Richardsonextrapolation zugeschnitten sind, und beide Techniken miteinander verglichen. Es zeigt sich, daß im Gegensatz zur üblichen Meinung Extrapolation nicht schlechter ist als Einbettung. Die numerischen Ergebnisse für die 25 Beispiele aus STIFF DETEST und einige schwierigere Beispiele zeigen folgendes Verhalten: Für niedrige Genauigkeiten (∼10−2) sind eingebettete Methoden vorteilhaft, für mittlere Genauigkeiten (∼10−4) sind beide Techniken gleichwertig und für hohe Genauigkeiten (<10−5) ist Extrapolation überlegen. Unter bestimmten Voraussetzungen kann der extrapolierte Wert als Startwert für den nächsten Schritt verwendet werden.

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Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • P. Kaps
    • 1
  • S. W. H. Poon
    • 2
  • T. D. Bui
    • 2
  1. 1.Institut für Mathematik und GeometrieUniversität InnsbruckInnsbruckAustria
  2. 2.Department of Computer ScienceConcordia UniversityMontréalCanada

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