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Abstract. In this paper a finite volume scheme for the heterogeneous and anisotropic diffusion equations is proposed on general, possibly nonconforming meshes. This scheme has both cell-centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear weighted combination of the surrounding cell-centered unknowns, which reduces the scheme to a completely cell-centered one. We propose two types of new explicit weights which allow arbitrary diffusion tensors, and are neither discontinuity dependent nor mesh topology dependent. Both the derivation of the scheme and that of new weights satisfy the linearity preserving criterion which requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is called as the linearity preserving cell-centered scheme and the numerical results show that it maintain optimal convergence rates for the solution and flux on general polygonal distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous.

We propose a new positivity-preserving finite volume scheme for the anisotropic diffusion problems on general polygonal meshes based on a new nonlinear two-point flux approximation. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are computed by two second-order interpolation algorithms. Due to the new nonlinear two-point flux formulation, it is not required to replace the interpolation algorithm with positivity-preserving but usually low-order accurate ones whenever negative interpolation weights occur and it is also unnecessary to require the decomposition of the conormal vector to be a convex one. Moreover, the new nonlinear two-point flux approximation has a fixed stencil. These features make our scheme more flexible, easy for implementation, and different from other existing nonlinear schemes based on Le Potier's two-point flux approximation. The positivity-preserving property of our scheme is proved theoretically and numerical results demonstrate that the scheme has nearly the same convergence rates as compared with other second-order accurate linear schemes, especially on severely distorted meshes.

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