We present a novel spatial discretization for the anisotropic heat conduction equation, aimed at improved accuracy at the high levels of anisotropy seen in a magnetized plasma, for example, for magnetic confinement fusion. The new discretization is based on a mixed formulation, introducing a form of the directional derivative along the magnetic field as an auxiliary variable and discretizing both the temperature and auxiliary fields in a continuous Galerkin (CG) space. Both the temperature and auxiliary variable equations are stabilized using the streamline upwind Petrov-Galerkin (SUPG) method, ensuring a better representation of the directional derivatives and therefore an overall more accurate solution. This approach can be seen as the CG-based version of our previous work (Wimmer, Southworth, Gregory, Tang, 2024), where we considered a mixed discontinuous Galerkin (DG) spatial discretization including DG-upwind stabilization. We prove consistency of the novel discretization, and demonstrate its improved accuracy over existing CG-based methods in test cases relevant to magnetic confinement fusion. This includes a long-run tokamak equilibrium sustainment scenario, demonstrating a 35% and 32% spurious heat loss for existing primal and mixed CG-based formulations versus 4% for our novel SUPG-stabilized discretization.

翻译：本文提出了一种针对各向异性热传导方程的新型空间离散化方法，旨在提高在磁化等离子体（例如磁约束聚变中）高各向异性水平下的计算精度。该离散化方法基于混合变分形式，通过引入沿磁场方向的方向导数作为辅助变量，并将温度场与辅助场均离散在连续伽辽金空间中。温度方程与辅助变量方程均采用流线迎风彼得罗夫-伽辽金方法进行稳定化处理，从而确保方向导数得到更精确的表示，最终获得整体精度更高的数值解。此方法可视为我们前期工作（Wimmer, Southworth, Gregory, Tang, 2024）的连续伽辽金版本，前期工作采用了包含间断伽辽金迎风稳定化的混合间断伽辽金空间离散化。我们证明了新离散格式的相容性，并通过磁约束聚变相关测试案例验证了其相较于现有连续伽辽金方法的精度提升。在长时托卡马克平衡维持场景中，现有基于原始形式与混合形式的连续伽辽金方法分别产生35%和32%的虚假热损失，而我们提出的SUPG稳定化离散格式仅产生4%的虚假热损失。