Asymptotic Preserving scheme for strongly anisotropic parabolic equations for arbitrary anisotropy direction

TL;DR

This paper introduces a new Asymptotic-Preserving numerical scheme for strongly anisotropic parabolic equations that remains accurate and efficient regardless of anisotropy strength or direction, applicable to fusion plasma simulations.

Contribution

It develops a novel Asymptotic-Preserving method that overcomes previous limitations related to anisotropy direction, maintaining accuracy without increased computational costs.

Findings

Method convergence is independent of anisotropy parameter
.

Applicable to arbitrary anisotropy directions with fixed coarse grids.

Effective for magnetically confined fusion plasma simulations.

Abstract

This paper deals with the numerical study of a strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. Furthermore, the recently proposed Asymptotic-Preserving method [arXiv:1203.6739] allows to perform simulations regardless of the anisotropy strength but its application is limited to the case, where the anisotropy direction is given by a field with all field lines open. In this paper we introduce a new Asymptotic-Preserving method, which overcomes those limitations without any loss of precision or increase in the computational costs. The convergence of the method is shown to be independent of the anisotropy parameter $0 < \eps <1$, and this for fixed coarse Cartesian grids and for variable anisotropy directions. The context of this work are magnetically confined fusion plasmas.