A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines

Wimmer, Golo A.; Southworth, Ben S.; Gregory, Thomas J.; Tang, Xian-Zhu

doi:10.1137/23m155918x

A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines

Journal Article · Fri May 24 00:00:00 EDT 2024 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/23m155918x· OSTI ID:2433925

^[1]; ^[1]; Gregory, Thomas J. ^[2]; ^[1]

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Imperial College, London (United Kingdom)

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 1000x smaller for anisotropy ratio of 10⁹, 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, which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are nonsingular, 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.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC), Fusion Energy Sciences (FES); USDOE Office of Science (SC). Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: 89233218CNA000001; AC02-05CH11231

OSTI ID:: 2433925

Report Number(s):: LA-UR--23-20885

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 3 Vol. 46; ISSN 1064-8275

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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