Convergence of alternant theta-method with applications

I. Faragó, Z. Farkas

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

Abstract

In this work the alternant theta-method and its application is investigated. We analyze the local approximation error and the convergence of the method on the non-equidistant mesh. We define the order of convergence, as well. The main idea of this approach is the approximation of the solution of the Cauchy problems by using different numerical schemes (implicit, explicit, IMEX, one-step, multi-step etc.) with varying step-sizes. Benefits of such approximations are shown for the problems with non-smooth solutions. We show that the convergence and the error estimation can be given relatively easily for the classical θ-method in case both equidistant and non-equidistant time/space discretizations. We analyze the connection of this approach to the classical discrete Gronwall lemma. We show that the extended discrete Gronwall lemma can be successfully applied to the estimation of the convergence’s rate of the alternant θi method. We show numerical examples for some non-linear time dependent differential equations, which have non-continuous or strongly oscillated solutions.

Original languageEnglish
Title of host publicationNumerical Analysis and Its Applications - 6th International Conference, NAA 2016, Revised Selected Papers
PublisherSpringer Verlag
Pages58-69
Number of pages12
ISBN (Print)9783319570983
DOIs
Publication statusPublished - Jan 1 2017
Event6th International Conference on Numerical Analysis and Its Applications, NAA 2016 - Lozenetz, Bulgaria
Duration: Jun 15 2016Jun 22 2016

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10187 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other6th International Conference on Numerical Analysis and Its Applications, NAA 2016
CountryBulgaria
CityLozenetz
Period6/15/166/22/16

Fingerprint

θ-method
Lemma
Error analysis
Differential equations
Local Approximation
Equidistant
Order of Convergence
Error Estimation
Approximation Error
Approximation
Numerical Scheme
Cauchy Problem
Rate of Convergence
Discretization
Mesh
Differential equation
Numerical Examples

Keywords

  • Alternant theta method
  • Gronwall lemma
  • Stability constant

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Faragó, I., & Farkas, Z. (2017). Convergence of alternant theta-method with applications. In Numerical Analysis and Its Applications - 6th International Conference, NAA 2016, Revised Selected Papers (pp. 58-69). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10187 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-57099-0_6

Convergence of alternant theta-method with applications. / Faragó, I.; Farkas, Z.

Numerical Analysis and Its Applications - 6th International Conference, NAA 2016, Revised Selected Papers. Springer Verlag, 2017. p. 58-69 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10187 LNCS).

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

Faragó, I & Farkas, Z 2017, Convergence of alternant theta-method with applications. in Numerical Analysis and Its Applications - 6th International Conference, NAA 2016, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10187 LNCS, Springer Verlag, pp. 58-69, 6th International Conference on Numerical Analysis and Its Applications, NAA 2016, Lozenetz, Bulgaria, 6/15/16. https://doi.org/10.1007/978-3-319-57099-0_6
Faragó I, Farkas Z. Convergence of alternant theta-method with applications. In Numerical Analysis and Its Applications - 6th International Conference, NAA 2016, Revised Selected Papers. Springer Verlag. 2017. p. 58-69. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-57099-0_6
Faragó, I. ; Farkas, Z. / Convergence of alternant theta-method with applications. Numerical Analysis and Its Applications - 6th International Conference, NAA 2016, Revised Selected Papers. Springer Verlag, 2017. pp. 58-69 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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