Abstract
Abstract Richardson Extrapolation is a very general numerical procedure, which can be applied in the solution of many mathematical problems in an attempt to increase the accuracy of the results. It is assumed that this approach is used to handle non-linear systems of ordinary differential equations (ODEs) which arise often in the mathematical description of scientific and engineering models either directly or after the discretization of the spatial derivatives of partial differential equations (PDEs). The major topic is the analysis of eight advanced implementations of the Richardson Extrapolation. Two important properties are analyzed: (a) the possibility to achieve more accurate results and (b) the possibility to improve the stability properties of eight advanced versions of the Richardson Extrapolation. A two-parameter family of test-examples was constructed and used to check both the accuracy and the absolute stability of the different versions of the Richardson Extrapolation when these versions are applied together with several Explicit Runge–Kutta Methods (ERKMs).
Highlights
It is well known that the Richardson Extrapolation is a very general approach, which can successfully be used during the approximate solution of different classes of mathematical problems in the efforts (a) to increase the accuracy of the selected numerical methods and/or (b) to control the stepsize when the treated mathematical problems are time-dependent
Explicit Runge–Kutta Methods with Richardson Extrapolation it is well known that the application of the Richardson Extrapolation in connection with time-depending problems is sometimes causing problems related to the absolute stability during the computational process ([9, 34, 36])
We shall assume that the order of accuracy p of the chosen Explicit Runge–Kutta Methods (ERKMs) is equal to the number m of the stage vectors kni, i = 1, 2, . . . , m
Summary
It is well known that the Richardson Extrapolation (introduced in [23], see [7, 32,33,34, 36, 37]) is a very general approach, which can successfully be used during the approximate solution of different classes of mathematical problems in the efforts (a) to increase the accuracy of the selected numerical methods and/or (b) to control the stepsize when the treated mathematical problems are time-dependent. We shall first introduce some convenient notations, which will after that be consistently used in this paper
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