Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit Runge-Kutta Methods

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

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

Abstract

Efficient implementation of the Two-times Repeated Richardson Extrapolation is studied in this paper under the assumption that systems of ordinary differential equations (ODEs) are solved numerically by Explicit Runge-Kutta Methods (ERKMs). The combinations of the Two-times Repeated Richardson Extrapolation with the ERKMs are new numerical methods. The computational cost per step of these new numerical methods is higher than the computational cost per step of the underlying ERKMs. However, the order of accuracy of the combined methods becomes very high: if the order of accuracy of the underlying ERKM is p, then the order of accuracy of its combination with the Two-times Repeated Richardson Extrapolation is at least p+3 when the right-hand-side function of the system of ODEs is sufficiently many times continuously differentiable. Moreover, the stability properties of the new methods are always better than those of the underlying numerical methods when p=m and m=1,2,3,4 (where m is the number of stage vectors in the chosen ERKM). These two useful properties, higher accuracy and better stability, are often giving a very reasonable compensation for the increased computational cost per step, because the same degree of accuracy can be achieved by applying a large stepsize which leads to a considerable reduction of the number of steps when the Two-times Repeated Richardson Extrapolation is used. This fact is verified by several numerical experiments.

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
Pages678-686
Number of pages9
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

Fingerprint

Richardson Extrapolation
Absolute Stability
Runge Kutta methods
Explicit Methods
Runge-Kutta Methods
Extrapolation
Computational Cost
Numerical methods
Numerical Methods
System of Ordinary Differential Equations
Ordinary differential equations
Costs
Combined Method
Continuously differentiable
Efficient Implementation
High Accuracy
Numerical Experiment
Experiments

Keywords

  • Absolute stability properties
  • Explicit Runge-Kutta Methods
  • Systems of ordinary differential equations (ODEs)
  • Two-times Repeated Richardson Extrapolation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Zlatev, Z., Dimov, I., Faragó, I., Georgiev, K., & Havasi, Á. (2019). Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit 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. 678-686). (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_80

Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit Runge-Kutta Methods. / Zlatev, Zahari; Dimov, Ivan; Faragó, I.; Georgiev, Krassimir; Havasi, Ágnes.

Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers. ed. / István Faragó; Ivan Dimov; Lubin Vulkov. Springer Verlag, 2019. p. 678-686 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11386 LNCS).

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

Zlatev, Z, Dimov, I, Faragó, I, Georgiev, K & Havasi, Á 2019, Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit 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. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11386 LNCS, Springer Verlag, pp. 678-686, 7th International Conference on Finite Difference Methods, FDM 2018, Lozenetz, Bulgaria, 6/11/18. https://doi.org/10.1007/978-3-030-11539-5_80
Zlatev Z, Dimov I, Faragó I, Georgiev K, Havasi Á. Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit Runge-Kutta Methods. In Faragó I, Dimov I, Vulkov L, editors, Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers. Springer Verlag. 2019. p. 678-686. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-030-11539-5_80
Zlatev, Zahari ; Dimov, Ivan ; Faragó, I. ; Georgiev, Krassimir ; Havasi, Ágnes. / Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit Runge-Kutta Methods. Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers. editor / István Faragó ; Ivan Dimov ; Lubin Vulkov. Springer Verlag, 2019. pp. 678-686 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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