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.
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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This work has been supported in part by the Deutsche Forschungsgemeinschaft and by the National Research Council of Canada.
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Kaps, P., Poon, S.W.H. & Bui, T.D. Rosenbrock methods for Stiff ODEs: A comparison of Richardson extrapolation and embedding technique. Computing 34, 17–40 (1985). https://doi.org/10.1007/BF02242171
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DOI: https://doi.org/10.1007/BF02242171