A fast algebraic multigrid solver and accurate discretization for highly anisotropic heat flux I: open field lines

TL;DR

This paper introduces a novel finite element discretization and an algebraic multigrid solver tailored for highly anisotropic heat flux equations in magnetic confinement fusion, achieving superior accuracy and efficiency especially for open field lines.

Contribution

It presents a new discretization method and an AMG solver approach specifically designed for anisotropic heat flux problems with open magnetic field lines, improving accuracy and convergence.

Findings

Achieves 1000x smaller error for high anisotropy ratios.

Demonstrates fast convergence in highly anisotropic regimes.

Superior accuracy over existing discretizations.

Abstract

We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. We simultaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators. We pose the problem in a mixed formulation, introducing the directional temperature gradient as an auxiliary variable. The temperature and auxiliary fields are discretized in a scalar discontinuous Galerkin space with upwinding principles used for discretizations of advection. We demonstrate the proposed discretization's superior accuracy over other discretizations of anisotropic heat flux, achieving error $1000\times$ smaller for anisotropy ratio of $10^9$, for $closed$ $field$ $lines$. The block matrix system is reordered and solved in an approach where the two advection operators are inverted using AMG solvers based on approximate ideal restriction (AIR), which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are non-singular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines for the linear solvers. We demonstrate fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.