NUMERICAL RESOLUTION OF AN ANISOTROPIC NON-LINEAR DIFFUSION PROBLEM *

Stéphane Brull (1) , Fabrice Deluzet (2) , Alexandre Mouton (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 LPP - Laboratoire Paul Painlevé - UMR 8524

This paper is devoted to the numerical resolution of an anisotropic non-linear diffusion problem involving a small parameter ǫ, defined as the anisotropy strength reciprocal. In this work, the anisotropy is carried by a variable vector function b. The equation being supplemented with Neumann boundary conditions, the limit ǫ → 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.

Mots clés: Anisotropic diffusion problems; Singular perturbation; Asymptotic-Preserving schemes AMS Subject Classification: 35J60; 35J62; 65M06; 65M12; 65N06; 65N12

Domaines: Analyse numérique [math.NA]

Stéphane Brull, Fabrice Deluzet, Alexandre Mouton. NUMERICAL RESOLUTION OF AN ANISOTROPIC NON-LINEAR DIFFUSION PROBLEM *. Communications in Mathematical Sciences, 2015. ⟨hal-02483867⟩