### 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 | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 1-11 |

Number of pages | 11 |

Volume | 8236 LNCS |

DOIs | |

Publication status | Published - 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) | 03029743 |

ISSN (Electronic) | 16113349 |

### 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 |

### Fingerprint

### 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

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Note on the convergence of the implicit Euler method

AU - Faragó, I.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - finite difference method

KW - implicit and explicit Euler method

KW - Numerical solution of ODE

KW - Runge-Kutta methods

UR - http://www.scopus.com/inward/record.url?scp=84886831251&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84886831251&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-41515-9_1

DO - 10.1007/978-3-642-41515-9_1

M3 - Conference contribution

AN - SCOPUS:84886831251

SN - 9783642415142

VL - 8236 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 11

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -