An improvement of the two-point flux approximation scheme on polygonal meshes

Highlights: • A diffusion scheme based on a nonlinear combination of two-side flux, instead of one-side flux, is constructed. • It is an improvement of the TPFA scheme by adding a nonlinear tangential component in the construction of discrete flux. • It satisfies the maximum principle and has a minimal stencil on polygonal meshes. • Numerical results show that it is second order accurate on 2D skewed meshes for anisotropic diffusion problems. -- Abstract: A new diffusion scheme satisfying the maximum principle on two-dimensional (2D) polygonal meshes is proposed. Being different from the existing schemes which are based on the nonlinear combination of the two one-side fluxes, the scheme in this paper is constructed by a modification of the conventional linear Two-Point Flux Approximation (TPFA) scheme, in which the two-side flux is constructed. To improve its accuracy on 2D skewed meshes, the TPFA scheme is rewritten by adding a tangential part, i.e., the tangential component of the flux multiplied with a nonlinear factor. A new method for the discretization of the tangential part is presented so that both the conservation condition and the maximum principle are satisfied, while the TPFA part remains unchanged. Moreover, by virtue of the harmonic averaging points for the calculation of the tangential part, the scheme has a minimal stencil, similar to the TPFA scheme. Finally, some numerical examples are presented to verify that it is second order accurate on 2D polygonal meshes for anisotropic diffusion problems.