### 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 language | English |
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Title of host publication | Numerical Analysis and Its Applications - 5th International Conference, NAA 2012, Revised Selected Papers |

Pages | 1-11 |

Number of pages | 11 |

DOIs | |

Publication status | Published - Nov 7 2013 |

Event | 5th International Conference on Numerical Analysis and Applications, NAA 2012 - Lozenetz, Bulgaria Duration: Jun 15 2013 → Jun 20 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8236 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 5th International Conference on Numerical Analysis and Applications, NAA 2012 |
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Country | Bulgaria |

City | Lozenetz |

Period | 6/15/13 → 6/20/13 |

### 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

*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