Numerical resolution of an anisotropic non-linear diffusion problem

TL;DR

This paper develops an Asymptotic-Preserving numerical scheme for solving a complex anisotropic non-linear diffusion problem with a small parameter, ensuring accuracy and efficiency regardless of anisotropy strength.

Contribution

The paper introduces a novel Asymptotic-Preserving scheme that handles anisotropic non-linear diffusion without coordinate adaptation, maintaining accuracy and computational efficiency.

Findings

Scheme is independent of anisotropy strength

Achieves high accuracy with low computational cost

Effectively handles singular perturbation limit

Abstract

This paper is devoted to the numerical resolution of an anisotropic non-linear diffusion problem involving a small parameter \varepsilon, 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 \varepsilon \infty 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.