Communications in Mathematical Sciences

Volume 13 (2015)

Number 1

Numerical resolution of an anisotropic non-linear diffusion problem

Pages: 203 – 224



Stéphane Brull (Institut de Mathématiques de Bordeaux, Université de Bordeaux 1, Talence, France)

Fabrice Deluzet (CNRS, Institut de Math´ematiques de Toulouse, France)

Alexandre Mouton (CNRS, Laboratoire Paul Painlevé, Université Sciences et Technologies de Lille, Villeneuve d’Ascq, France)


This paper is devoted to the numerical resolution of an anisotropic non-linear diffusion problem involving a small parameter $\epsilon$, defined as the anisotropy strength reciprocal. In this work, the anisotropy is carried by a variable vector function $\mathbf{b}$. The equation being supplemented with Neumann boundary conditions, the limit $\epsilon \to 0$ is demonstrated to be a singular perturbation of the original diffusion equation. To address efficiently this problem, an Asymptotic-Preserving scheme is derived. This numerical method does not require the use of coordinates adapted to the anisotropy direction and exhibits an accuracy as well as a computational cost independent of the anisotropy strength.


anisotropic diffusion problems, singular perturbation, asymptotic-preserving schemes

2010 Mathematics Subject Classification

35J60, 35J62, 65M06, 65M12, 65N06, 65N12

Published 16 July 2014