Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-29T08:02:50.361Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 January 2003

Florian Mehats
Florian Mehats*
Affiliation:
MIP, UMR CNRS 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 04, France. [email protected].
Get access

Abstract

We present here a discretization of a nonlinear obliquederivative boundary value problem for the heat equation in dimensiontwo. This finite difference scheme takes advantages of thestructure of the boundary condition, which can be reinterpreted as aBurgers equation in the space variables. This enables to obtain anenergy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of thisproblem and a numerical study of the stability of the scheme, whichappears to be in good agreement with the theory.

Keywords

Oblique derivative boundary problemfinite difference schemeheat equationBurgers equation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 6 , November 2002 , pp. 1111 - 1132
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Caffarelli, L. and Roquejoffre, J.-M., 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. CrossRef
Dong, G., Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations. J. Partial Differential Equations Ser. A 1 (1988) 12-42.
E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Mathématiques & Applications, Ellipse, Paris (1991).
B. Larrouturou, Modélisation mathématique et numérique pour les sciences de l'ingénieur. Cours de l'École polytechnique, Département de Mathématiques Appliquées, 1996.
R.J. LeVeque, Numerical Methods for Conservation Laws. Lectures in Mathematics, Birkhäuser Verlag (1990).
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Études Mathématiques, Dunod, Gauthier-Villars (1969).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et recherches Mathématiques, Dunod (1968).
F. Méhats, Étude de problèmes aux limites en physique du transport des particules chargées. Thèse de doctorat (1997).
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.
Nazarov, A.I. and Ural'tseva, N.N., Problem, A with an Oblique Derivative for a Quasilinear Parabolic Equation. J. Math. Sci. 77 (1995) 3212-3220. CrossRef