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 publicationNumerical Analysis and Its Applications - 5th International Conference, NAA 2012, Revised Selected Papers
Pages1-11
Number of pages11
DOIs
Publication statusPublished - Nov 7 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)0302-9743
ISSN (Electronic)1611-3349

Other

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

Keywords

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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Faragó, I. (2013). Note on the convergence of the implicit Euler method. In Numerical Analysis and Its Applications - 5th International Conference, NAA 2012, Revised Selected Papers (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