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.

翻译：摘要：本文针对磁约束聚变等离子体中典型的高各向异性热通量问题，提出了一种新型求解技术。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。我们通过构建兼具优异精度特性的有限元离散格式，并针对基于对流算子设计的代数多重网格（AMG）方法定制新型求解策略，同步解决了上述难题。我们采用混合形式建模，引入方向温度梯度作为辅助变量，并采用含迎风离散机制的标量间断伽辽金空间对温度场与辅助场进行离散化。在封闭场线条件下，当各向异性比达$10^9$时，所提离散化方案的误差较其他各向异性热通量离散方法降低$1000$倍。通过重排序块矩阵系统，我们采用基于近似理想限制（AIR）的AMG求解器对两个对流算子进行求逆，该技术对迎风间断伽辽金离散化具有极高效率。为确保对流算子非奇异，本文在求解器设计中限定于开放（非循环）磁力线场景。我们证明，在基于扩散的AMG方法失效的高各向异性区域，所提迭代求解器仍能实现快速收敛。