We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. 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. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. A nonlinear penalty parameter is shown to provide an effective method for tuning the parallel diffusion penalty and significantly minimises rounding errors. Several numerical experiments, using manufactured solutions, the ``NIMROD benchmark'' problem and a single island problem, are presented to verify numerical accuracy, convergence, and asymptotic preserving properties of the method. 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：我们提出了一种新颖的数值方法，用于求解周期性盒子中受限磁场下的各向异性扩散方程。该方法具有高精度且可证明是稳定的。我们推导了连续初始边值问题解的能量估计。采用时间算子分裂技术，并结合求和分部有限差分近似处理垂直扩散算子的空间导数，建立了离散公式。通过场线追踪，推导了实现边界条件和平行扩散算子的弱惩罚过程。我们证明全离散逼近是无条件稳定的。在正确选择惩罚参数的情况下，离散能量估计与连续能量估计相匹配。研究表明，非线性惩罚参数能有效调节平行扩散惩罚项，并显著减少舍入误差。通过使用制造解、"NIMROD基准"问题以及单岛问题的多个数值实验，验证了该方法的数值精度、收敛性和渐近保持特性。最后，我们展示了包含混沌区域和岛的磁场，并表明各向异性扩散方程的等值线再现了场中的关键特征。