Note on the convergence of the implicit Euler method

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

1 Citation (Scopus)

Abstract

For the solution of the Cauchy problem for the first order ODE, the most popular, simplest and widely used method are the Euler methods. The two basic variants of the Euler methods are the explicit Euler methods (EEM) and the implicit Euler method (IEM). These methods are well-known and they are introduced almost in any arbitrary textbook of the numerical analysis, and their consistency is given. However, in the investigation of these methods there is a difference in concerning the convergence: for the EEM it is done almost everywhere but for the IEM usually it is missed. (E.g., [1, 2, 6-9].)The stability (and hence, the convergence) property of the IEM is usually shown as a consequence of some more general theory. Typically, from the theory for the implicit Runge-Kutta methods, which requires knowledge of several basic notions in numerical analysis of ODE theory, and the proofs are rather complicated. In this communication we will present an easy and elementary prove for the convergence of the IEM for the scalar ODE problem. This proof is direct and it is available for the non-specialists, too.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Pages1-11
Number of pages11
Volume8236 LNCS
DOIs
Publication statusPublished - 2013
Event5th International Conference on Numerical Analysis and Applications, NAA 2012 - Lozenetz, Bulgaria
Duration: Jun 15 2013Jun 20 2013

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8236 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other5th International Conference on Numerical Analysis and Applications, NAA 2012
CountryBulgaria
CityLozenetz
Period6/15/136/20/13

Fingerprint

Euler's method
Numerical analysis
Runge Kutta methods
Textbooks
Communication
Numerical Analysis
Implicit Runge-Kutta Methods
Convergence Properties
Cauchy Problem
Scalar
First-order
Arbitrary

Keywords

  • finite difference method
  • implicit and explicit Euler method
  • Numerical solution of ODE
  • Runge-Kutta methods

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Faragó, I. (2013). Note on the convergence of the implicit Euler method. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8236 LNCS, pp. 1-11). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8236 LNCS). https://doi.org/10.1007/978-3-642-41515-9_1

Note on the convergence of the implicit Euler method. / Faragó, I.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 8236 LNCS 2013. p. 1-11 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8236 LNCS).

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

Faragó, I 2013, Note on the convergence of the implicit Euler method. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 8236 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8236 LNCS, pp. 1-11, 5th International Conference on Numerical Analysis and Applications, NAA 2012, Lozenetz, Bulgaria, 6/15/13. https://doi.org/10.1007/978-3-642-41515-9_1
Faragó I. Note on the convergence of the implicit Euler method. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 8236 LNCS. 2013. p. 1-11. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-41515-9_1
Faragó, I. / Note on the convergence of the implicit Euler method. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 8236 LNCS 2013. pp. 1-11 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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