A hyperbolic finite difference scheme for anisotropic diffusion equations: Preserving the discrete maximum principle

The hyperbolic system approach is a promising method for robust computation of anisotropic diffusion equations that arise in quasineutral plasmas. Though the approach exhibits merits of high extensibility and accurate flux computation, the characteristics of the scheme in terms of the discrete maximum principle (DMP) for anisotropic diffusion cases has not been understood. In this study, the DMP of the hyperbolic system approach is analyzed and tested in various anisotropic diffusion cases, with particular attention to the role of an arbitrary parameter that appears in the scheme. A mathematical analysis is conducted to obtain an optimal condition of the arbitrary parameter to guarantee the DMP, and numerical experiments reveal an adoptive selection of the parameter for DMP-preserving results. It is confirmed that, with an appropriate preconditioning matrix and parameter choice, the hyperbolic system approach preserves the DMP even with a linear discretization.