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

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.