A Maximum-Principle-Preserving Finite Volume Scheme for Diffusion Problems on Distorted Meshes

Abstract

In this paper, we propose an approach for constructing conservative and maximum-principle-preserving finite volume schemes by using the method of undetermined coefficients, which depend nonlinearly on the linear non-conservative one-sided fluxes. In order to facilitate the derivation of expressions of these undetermined coefficients, we explicitly provide a simple constriction condition with a scaling parameter. Such constriction conditions can ensure the final schemes are exact for linear solution problems and may induce various schemes by choosing different values for the parameter. In particular, when this parameter is taken to be 0, the nonlinear terms in our scheme degenerate to a harmonic average combination of the discrete linear fluxes, which has often been used in a variety of maximum-principle-preserving finite volume schemes. Thus our method of determining the coefficients of the nonlinear terms is more general. In addition, we prove the convergence of the proposed schemes by using a compactness technique. Numerical results demonstrate that our schemes can preserve the conservation property, satisfy the discrete maximum principle, possess a second-order accuracy, be exact for linear solution problems, and be available for anisotropic problems on distorted meshes.