On quasi-consistent integration by Nordsieck methods
type: Item
instance of: scholarly article
title: On quasi-consistent integration by Nordsieck methods
zbMATH Open document ID: 1163.65058
author: G. Yu. Kulikov
published in: Journal of Computational and Applied Mathematics
publication date: 2009-03-02T00:00:00Z
review text: Quasi-consistent numerical integration of ordinary differential equations (ODEs) \(x'(t)=g(t,x(t))\) is available since 1976, but few use in practical applications has been done, mainly because of difficulties in the implementation such as the problem of quasi-consistent integration in a non-equidistant grid. The author studies the advantages of numerical integration by means of quasi-consistent Nordsieck formulae. He exploits the idea that the global error control can be as cheap as the local error control if the principal term of the local error dominates strongly over the remaining terms (the accumulated error does not exceed the principal term of the local error, being both terms of the same order with respect to the step-size) and considers Nordsieck-Adams-Moulton methods for an efficient global error control. The author also introduces implicitly extended Nordsieck methods to facilitate the local error control and studies the convergence of these methods. Moreover the paper ends with quite a few numerical examples using implicitly extended Nordsieck-Adams-Moulton formulae of orders from 3 to 7. One of the main theoretical achievements is the introduction and study of doubly quasi-consistent methods. Methods of this type accumulate the errors of two order higher than the local error of the numerical solution, meaning that the principal terms of local and global errors of a doubly quasi-consistent method coincide, and so the principal term of the local error is an asymptotically correct estimate of the true error. Unfortunately, as it is proved in the paper, Nordsieck schemes cannot be doubly quasi-consistent and although the idea of doubly quasi-consistent methods looks promising, and the theoretical background for this new global error evaluation is provided, no one has been found yet. Only fixed-step-size Nordsieck formulae are exploited in the paper, however using the idea that the local error can be a good estimate of the global one in super-quasi-consisten…
reviewed by: Josep J. Masdemont
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 5522055
zbMATH Keywords: quasi-consistent Nordsieck methods
zbMATH Keywords: double quasi-consistency
zbMATH Keywords: implicitly extended Nordsieck formulas
zbMATH Keywords: adaptivity
zbMATH Keywords: local error estimation
zbMATH Keywords: automatic global error control
zbMATH Keywords: Nordsieck-Adams-Moulton methods
zbMATH Keywords: numerical examples
describes a project that uses: GEAR
describes a project that uses: DIFSUB
MaRDI profile type: Publication
full work available at URL: j.cam.2008.07.038
OpenAlex ID: W2168204527
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DOI: 10.1016/J.CAM.2008.07.038
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description: scientific article; zbMATH DE number 5522055