An implicit scheme to solve a system of ODEs arising from the space discretization of nonlinear diffusion equations

Éric Boillat

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 4, page 749-765
  • ISSN: 0764-583X

Abstract

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In this article, we consider the initial value problem which is obtained after a space discretization (with space step h ) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size h chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between h and the time step size τ . Moreover, it is not of excessive algorithmic complexity since it does not require more than one resolution of a linear system at each time step.

How to cite

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Boillat, Éric. "An implicit scheme to solve a system of ODEs arising from the space discretization of nonlinear diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.4 (2001): 749-765. <http://eudml.org/doc/194071>.

@article{Boillat2001,
abstract = {In this article, we consider the initial value problem which is obtained after a space discretization (with space step $h$) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size $h$ chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between $h$ and the time step size $\tau $. Moreover, it is not of excessive algorithmic complexity since it does not require more than one resolution of a linear system at each time step.},
author = {Boillat, Éric},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {nonlinear diffusion equations; nonlinear parabolic problem; Chernoff scheme; implicit scheme for ODE’s; nonlinear parabolic problems; diffusion equations; initial value problem; solidification; multicomponent alloy; global convergence; stability; algorithmic complexity; Chernoff algorithm},
language = {eng},
number = {4},
pages = {749-765},
publisher = {EDP-Sciences},
title = {An implicit scheme to solve a system of ODEs arising from the space discretization of nonlinear diffusion equations},
url = {http://eudml.org/doc/194071},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Boillat, Éric
TI - An implicit scheme to solve a system of ODEs arising from the space discretization of nonlinear diffusion equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 4
SP - 749
EP - 765
AB - In this article, we consider the initial value problem which is obtained after a space discretization (with space step $h$) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size $h$ chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between $h$ and the time step size $\tau $. Moreover, it is not of excessive algorithmic complexity since it does not require more than one resolution of a linear system at each time step.
LA - eng
KW - nonlinear diffusion equations; nonlinear parabolic problem; Chernoff scheme; implicit scheme for ODE’s; nonlinear parabolic problems; diffusion equations; initial value problem; solidification; multicomponent alloy; global convergence; stability; algorithmic complexity; Chernoff algorithm
UR - http://eudml.org/doc/194071
ER -

References

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