# Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem

- Volume: 36, Issue: 6, page 1111-1132
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topMehats, Florian. "Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.6 (2002): 1111-1132. <http://eudml.org/doc/245936>.

@article{Mehats2002,

abstract = {We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears to be in good agreement with the theory.},

author = {Mehats, Florian},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {oblique derivative boundary problem; finite difference scheme; heat equation; Burgers equation},

language = {eng},

number = {6},

pages = {1111-1132},

publisher = {EDP-Sciences},

title = {Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem},

url = {http://eudml.org/doc/245936},

volume = {36},

year = {2002},

}

TY - JOUR

AU - Mehats, Florian

TI - Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 6

SP - 1111

EP - 1132

AB - We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears to be in good agreement with the theory.

LA - eng

KW - oblique derivative boundary problem; finite difference scheme; heat equation; Burgers equation

UR - http://eudml.org/doc/245936

ER -

## References

top- [1] L. Caffarelli and J.-M. Roquejoffre, A nonlinear oblique derivative boundary value problem for the heat equation: analogy with the porous medium equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 41–80. Zbl1016.35038
- [2] G. Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations. J. Partial Differential Equations Ser. A 1 (1988) 12–42. Zbl0699.35152
- [3] E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Mathématiques & Applications, Ellipse, Paris (1991). Zbl0768.35059MR1304494
- [4] B. Larrouturou, Modélisation mathématique et numérique pour les sciences de l’ingénieur. Cours de l’École polytechnique, Département de Mathématiques Appliquées, 1996.
- [5] R.J. LeVeque, Numerical Methods for Conservation Laws. Lectures in Mathematics, Birkhäuser Verlag (1990). Zbl0723.65067MR1077828
- [6] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Études Mathématiques, Dunod, Gauthier-Villars (1969). Zbl0189.40603MR259693
- [7] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et recherches Mathématiques, Dunod (1968). Zbl0165.10801MR247243
- [8] F. Méhats, Étude de problèmes aux limites en physique du transport des particules chargées. Thèse de doctorat (1997).
- [9] F. Méhats and J.-M. Roquejoffre, A nonlinear oblique derivative boundary value problem for the heat equation, Part 1 and Part 2. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 221–253 and 691–724. Zbl0922.35072
- [10] A.I. Nazarov and N.N. Ural’tseva, A Problem with an Oblique Derivative for a Quasilinear Parabolic Equation. J. Math. Sci. 77 (1995) 3212–3220. Zbl0836.35075

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.