A robust, interpolation-free and monotone finite volume scheme for diffusion equations on arbitrary quadrilateral meshes

A novel monotone finite volume (FV) scheme on arbitrary quadrilateral meshes, such as extremely distorted meshes with concave, twisted, or even negative volume cells, is proposed to solve diffusion equations with discontinuous tensor coefficients. The first advantage of our scheme is that no auxiliary unknown is needed thus the efficiency and accuracy are improved. The second advantage is that the convex decomposition of the conormal vector relies only on the locations of cell centers and reduces geometric restrictions on cell shapes to preserve positivity. The difficulties of constructing discrete normal fluxes across the material interface are overcome by introducing primary unknowns on the opposite side of the interface into the discrete stencil with some virtual adjustments on their positions, based on the continuity of normal flux and tangent gradients. Numerical tests demonstrate that our new scheme is second-order accurate, unconditionally monotone, and comparisons with some existing methods in terms of efficiency and accuracy are also given.