A reconstruction algorithm with flexible stencils for anisotropic diffusion equations on 2D skewed meshes

A new reconstruction algorithm is developed to obtain diffusion schemes with cell-centered unknowns only. The main characteristic of the new algorithm is the flexibility of stencils when the auxiliary unknowns are reconstructed with cell-centered unknowns. The stencils are selected depending on the mesh geometry and discontinuities of diffusion coefficients. Moreover, an explicit expression is derived for interpolating the auxiliary unknowns in terms of cell-centered unknowns, and the auxiliary unknowns can be defined at any point on the edge. The algorithm is applied to construct several new diffusion schemes, whose effectiveness is illustrated by numerical experiments. For anisotropic problems with or without discontinuities, nearly second order accuracy is achieved on skewed meshes.