Tailored finite point method for the approximation of diffusion operators with non-symmetric diffusion tensor

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. (C) 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.