A vertex-centered finite volume scheme preserving the discrete maximum principle for anisotropic and discontinuous diffusion equations

Abstract In this paper, we propose one nonlinear vertex-centered finite volume (FV) scheme preserving the discrete maximum principle (DMP) for diffusion equations on distorted mildly meshes. We construct the vertex-centered scheme on dual meshes. By using harmonic average, we get the conservation of scheme. To satisfy the DMP, some existing schemes have to impose severe restrictions on the diffusion coefficient and meshes, which include that only one discontinuity is allowed in the domain. Our scheme removes these restrictions and can deal with the problem with any discontinuity on primary cell edges. Besides, our vertex-centered scheme does not involve the harmonic average of diffusion coefficients in different cells, which is remarkably distinguishable from cell-centered schemes, hence it does not suffer the numerical heat-barrier issue. Numerical results show that our scheme obtains second-order accuracy for continuous problems and those problems with more than one discontinuity. Moreover, our scheme can obtain much better accuracy and computational efficiency than other existing DMP-preserving schemes. At the same time, our scheme can handle the anisotropic problems and satisfies the DMP.