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：我们提出了一种高效、精确且可证明稳定的新型数值方法，用于求解环向约束磁场中的各向异性扩散方程。连续问题通过垂直于输运的导数算子和平行于输运的线性算子（通过磁力线追踪获得）进行表述。我们推导了连续初边值问题解的能量估计。采用时间算子分裂方法，结合垂直于扩散算子的空间导数求和分部有限差分逼近，提出了离散格式。推导了弱惩罚过程，用于实现边界条件和通过磁力线追踪获得的平行扩散算子。我们证明了全离散逼近是无条件稳定且渐近保持的。当正确选择惩罚参数时，离散能量估计与连续能量估计相匹配。针对垂直于算子单独进行了收敛性测试，并采用"NIMROD基准"问题作为制造解，展示即使在垂直于扩散为零的情况下，完整格式也能收敛。最后，我们给出了具有混沌区域和磁岛的磁场分布，并展示了各向异性扩散方程的解等高线重现了磁场中的关键特征。