A Maximum-Principle-Preserving Finite Volume Scheme for Diffusion Problems on Distorted Meshes (Advances in Applied Mathematics and Mechanics, 2023)
In this paper, we propose an approach for constructing conservative andmaximum-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 undeterminedcoefficients, we explicitly provide a simple constriction condition with a scaling parameter. Such constriction conditions can ensure the final schemes are exact for linearsolution problems and may induce various schemes by choosing different values forthe parameter. In particular, when this parameter is taken to be 0, the nonlinear termsin our scheme degenerate to a harmonic average combination of the discrete linearfluxes, which has often been used in a variety of maximum-principle-preserving finitevolume schemes. Thus our method of determining the coefficients of the nonlinearterms is more general. In addition, we prove the convergence of the proposed schemesby using a compactness technique. Numerical results demonstrate that our schemescan preserve the conservation property, satisfy the discrete maximum principle, possess a second-order accuracy, be exact for linear solution problems, and be availablefor anisotropic problems on distorted meshes.