A finite volume scheme preserving maximum principle with cell-centered and vertex unknowns for diffusion equations on distorted meshes

• Published in: Applied mathematics and computation Vol. 398; p. 125989
• Main Authors: Wang, Jiangfu, Sheng, Zhiqiang, Yuan, Guangwei
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.06.2021
• Subjects: Cell-centered unknowns; Discrete maximum principle; Distorted meshes; Finite volume scheme; Vertex unknowns; Finite volume scheme; Discrete maximum principle; Distorted meshes; Cell-centered unknowns; Vertex unknowns
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2021.125989

•The scheme regards the cell-centered unknowns and vertex unknowns as the primary un-knowns.•The scheme obtains second-order accuracy for solution and first-order accuracy for flux, and it’s accuracy is similar to and even better than that of existing schemes.•The scheme can deal with the diffusion problems with two or more discontinuous lines, which some existing schemes can not work for.•The scheme can solve anisotropic and heterogeneous problems.•The scheme satisfies the discrete maximum principle. In this paper, we propose a new nonlinear finite volume (FV) scheme preserving the discrete maximum principle (DMP) for diffusion equations on distorted meshes. We introduce both cell-centered and vertex unknowns as primary ones. It is well-known that some restrictions on the diffusion coefficients and meshes have to be imposed for existing cell-centered schemes to preserve the DMP (see, e.g., Sheng and Yuan 2011,2018), while our scheme here removes these restrictions and can apply to diffusion problems with arbitrary discontinuity on general meshes. That is, the new scheme avoids failure of handling arbitrary discontinuity on general meshes for existing cell-centered schemes, moreover it is proved that our scheme satisfies the DMP. Numerical results show that our scheme can obtain second-order accuracy, deal with the problems with anisotropic and heterogeneous diffusion coefficient and satisfy the DMP. In addition, it is verified that our scheme can compete with some existing schemes, even behaves better in regard to numerical accuracy and efficiency.