We present a novel numerical method for solving the anisotropic diffusion equation in toroidally confined magnetic fields which is efficient, accurate and provably stable. The continuous problem is written in terms of a derivative operator for the perpendicular transport and a linear operator, obtained through field line tracing, for the parallel transport. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable and asymptotic preserving. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. Convergence tests are shown for the perpendicular operator by itself, and the ``NIMROD benchmark" problem is used as a manufactured solution to show the full scheme converges even in the case where the perpendicular diffusion is zero. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.

翻译：本文提出一种高效、精确且可证明稳定的新型数值方法，用于求解环面约束磁场中的各向异性扩散方程。连续问题通过垂直输运的导数算子和磁场线追踪获得的平行输运线性算子进行表述，推导了连续初边值问题解的能量估计。采用时间算子分裂法构建离散格式，其中垂直扩散算子空间导数采用求和分部有限差分近似。通过弱罚函数方法实现了边界条件与磁场线追踪所得平行扩散算子的处理。证明全离散近似具有无条件稳定性和渐近保持特性，在罚参数恰当选择下离散能量估计与连续能量估计一致。分别对垂直算子进行收敛性测试，并采用"NIMROD基准"问题的制造解验证了完整格式在垂直扩散为零时的收敛性。最后通过含混沌区域和磁岛结构的磁场模型，展示了各向异性扩散方程等值线再现磁场关键特征的能力。