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Erschienen in:
2022 | OriginalPaper | Buchkapitel
On the Consistency Order of Runge–Kutta Methods Combined with Active Richardson Extrapolation
verfasst von : Teshome Bayleyegn, István Faragó, Ágnes Havasi
Erschienen in: Large-Scale Scientific Computing
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Abstract
Passive and active Richardson extrapolations are robust devices to increase the rate of convergence of time integration methods. While the order of convergence is shown to increase by one under rather natural smoothness conditions if the passive Richardson extrapolation is used, for the active Richardson extrapolation the increase of the order has not been generally proven. It is known that the Lipschitz property of the right-hand side function of the differential equation to be solved yields convergence of order p if the method is consistent in order p. In this paper it is shown that the active Richardson extrapolation increases the order of consistency by one when the underlying method is any Runge–Kutta method of order \(p=1,2\), or 3.