A family of linearity-preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids

A family of cell-centered finite volume schemes are proposed for anisotropic diffusion problems on arbitrary polyhedral grids with planar facets. The derivation of the schemes is done under a general framework through a certain linearity-preserving approach. The key ingredient of our algorithm is to employ solely the so-called harmonic averaging points located at the cell interfaces to define the auxiliary unknowns, which not only makes the interpolation procedure for auxiliary unknowns simple and positivity-preserving, but also reduces the stencil of the schemes. The final schemes are cell-centered with a small stencil of 25-point on the structured hexahedral grids. Moreover, the schemes satisfy the local conservation condition, treat discontinuity exactly and allow for a simple stability analysis. A second-order accuracy in the L 2 norm and a first-order accuracy in the H 1 norm are observed numerically on general distorted meshes in case that the diffusion tensor is anisotropic and discontinuous.