A nine-point scheme with explicit weights for diffusion equations on distorted meshes

In this paper, for the structured quadrilateral mesh we derive a nine-point difference scheme which has five cell-centered unknowns and four vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear combination of the neighboring cell-centered unknowns, which reduces the scheme to a cell-centered one with a local stencil involving nine cell-centered unknowns. The coefficients in the linear combination are known as the weights and two types of new weights are proposed. These new weights are neither discontinuity dependent nor mesh topology dependent, have explicit expressions, can reduce to the one-dimensional harmonic-average weights on the nonuniform rectangular meshes, and moreover, are easily extended to the unstructured polygonal meshes and non-matching meshes. Both the derivation of the nine-point scheme and that of new weights satisfy the linearity preserving criterion. Numerical experiments show that, with these new weights, the nine-point difference scheme and its simple extension have a nearly second order accuracy on many highly distorted meshes, including structured quadrilateral meshes, unstructured polygonal meshes and non-matching meshes.