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

Florian Mehats

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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.

How to cite

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Mehats, Florian. "Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem." ESAIM: Mathematical Modelling and Numerical Analysis 36.6 (2010): 1111-1132. <http://eudml.org/doc/194142>.

@article{Mehats2010,
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},
keywords = {Oblique derivative boundary problem; finite difference scheme; heat equation; Burgers equation.; finite difference scheme; Burgers equation},
language = {eng},
month = {3},
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/194142},
volume = {36},
year = {2010},
}

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
DA - 2010/3//
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.; finite difference scheme; Burgers equation
UR - http://eudml.org/doc/194142
ER -

References

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  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 Linéaire19 (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. A1 (1988) 12-42.  Zbl0699.35152
  3. E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Mathématiques & Applications, Ellipse, Paris (1991).  Zbl0768.35059
  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).  Zbl0723.65067
  6. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Études Mathématiques, Dunod, Gauthier-Villars (1969).  Zbl0189.40603
  7. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et recherches Mathématiques, Dunod (1968).  Zbl0165.10801
  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 Linéaire16 (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.  

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