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Note on the Convergence of the Implicit Euler Method

  • István Faragó
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

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.

Keywords

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • István Faragó
    • 1
  1. 1.Institute of Mathematics and MTA-ELTE “Numerical Analysis and Large Networks” Research GroupEötvös Loránd UniversityBudapestHungary

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