Abstract We present a tailored finite point method (TFPM) for anisotropic diffusion equations with a non-symmetric diffusion tensor on Cartesian grids. The fluxes on each edge are discretized by using a linear combination of the local basis functions, which come from the exact solution of the diffusion equation with constant coefficients on the local cell. In this way, the scheme is fully consistent and the flux is naturally continuous across the interfaces between the subdomains with a non-symmetric diffusion tensor. Additionally, it is convenient to handle the Neumann boundary condition or a variant of Neumann boundary condition. Numerical results obtained from solving different anisotropic diffusion problems including problems with sharp discontinuity near the interface and boundary, show that this approach is efficient. Second order convergence rate can be obtained with the numerical examples. The new scheme is also tested on a time-dependent problem modeling the Hall effect in the resistive magnetohydrodynamics, the results further show the robustness of this new method.

中文翻译：

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用非对称扩散张量逼近扩散算子的定制有限点方法

摘要 我们针对笛卡尔网格上具有非对称扩散张量的各向异性扩散方程提出了一种定制的有限点方法 (TFPM)。每条边上的通量是通过使用局部基函数的线性组合离散化的，局部基函数来自局部单元上具有常数系数的扩散方程的精确解。通过这种方式，该方案是完全一致的，并且通量在具有非对称扩散张量的子域之间的界面上自然是连续的。此外，可以方便地处理诺依曼边界条件或诺依曼边界条件的变体。从解决不同的各向异性扩散问题（包括界面和边界附近的明显不连续性问题）获得的数值结果表明，这种方法是有效的。通过数值例子可以得到二阶收敛速度。新方案还在电阻磁流体动力学中的霍尔效应建模的瞬态问题上进行了测试，结果进一步表明了这种新方法的稳健性。