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 heat flux as an auxiliary variable and discretizing the temperature and auxiliary fields in a discontinuous Galerkin space. The resulting block matrix system is then reordered and solved using an approach in which 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. We demonstrate the proposed discretization's superior accuracy over other discretizations of anisotropic heat flux, achieving error $1000\times$ smaller for anisotropy ratio of $10^9$, while also demonstrating fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.

翻译：我们针对磁约束聚变等离子体中常见的高度各向异性热通量方程，提出了一种新颖的求解器技术。此类问题面临两大挑战：(i) 离散化精度与(ii) 高效隐式线性求解器。我们通过构建具有优异精度特性的新有限元离散化，并针对基于代数多重网格（AMG）方法的对流算子求解器进行定制，同时应对了这两项挑战。该方法采用混合形式，将热通量作为辅助变量引入，并在间断伽辽金空间中对温度场与辅助场进行离散化。随后对所得块矩阵系统进行重排序，并利用基于近似理想限制（AIR）的AMG求解器反演两个对流算子，这种策略尤其适用于对流问题的迎风间断伽辽金离散化。为确保对流算子非奇异，本文仅考虑开放（非闭合）磁力线情形。我们证明了所提离散化在高度各向异性热通量计算中具有优于其他离散化方法的精度：当各向异性比达到10^9时，误差降低1000倍；同时，在现有扩散型AMG方法失效的高度各向异性区域中，该迭代求解器仍能实现快速收敛。