Stability Properties of Repeated Richardson Extrapolation Applied Together with Some Implicit Runge-Kutta Methods

Zahari Zlatev, Ivan Dimov, I. Faragó, Krassimir Georgiev, Ágnes Havasi

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Repeated Richardson Extrapolation can successfully be used in the efforts to improve the efficiency of the numerical treatment of systems of ordinary differential equations (ODEs) mainly by increasing the accuracy of the computed results. It is assumed in this paper that Implicit Runge-Kutta Methods (IRKMs) are used in the numerical solution of systems of ODEs. If the order of accuracy of the selected IRKM is p, then the order of accuracy of its combination with the Repeated Richardson Extrapolation is at least p+2 (assuming here that the right-hand-side of the system of ODEs is sufficiently many times continuously differentiable). However, it is additionally necessary to establish that the absolute stability properties of the new numerical methods (that are combinations of the Repeated Richardson Extrapolation and the selected IRKMs) are preserved, and this is an extremely difficult problem. Results related to the stability of the computations are derived and numerical tests with a two-parameter system of three ODEs and an atmospheric chemical scheme with 56 compounds, which is defined mathematically by a very stiff and ill-conditioned system of non-linear ODEs, are presented. The research results described in this paper can be considered as a continuation of the study carried out in Zlatev et al.: Richardson Extrapolation: Practical Aspects and Applications. De Gruyter, Berlin (2017).

Original languageEnglish
Title of host publicationFinite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers
EditorsIstván Faragó, Ivan Dimov, Lubin Vulkov
PublisherSpringer Verlag
Pages114-125
Number of pages12
ISBN (Print)9783030115388
DOIs
Publication statusPublished - Jan 1 2019
Event7th International Conference on Finite Difference Methods, FDM 2018 - Lozenetz, Bulgaria
Duration: Jun 11 2018Jun 16 2018

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11386 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference7th International Conference on Finite Difference Methods, FDM 2018
CountryBulgaria
CityLozenetz
Period6/11/186/16/18

Keywords

  • Absolute stability properties
  • Atmospheric chemical schemes
  • Implicit Runge-Kutta Methods
  • Repeated Richardson Extrapolation
  • Systems of ordinary differential equations (ODEs)

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Zlatev, Z., Dimov, I., Faragó, I., Georgiev, K., & Havasi, Á. (2019). Stability Properties of Repeated Richardson Extrapolation Applied Together with Some Implicit Runge-Kutta Methods. In I. Faragó, I. Dimov, & L. Vulkov (Eds.), Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers (pp. 114-125). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11386 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-11539-5_11