Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids

We propose two interpolation-based monotone schemes for the anisotropic diffusion problems on unstructured polygonal meshes through the linearity-preserving approach. The new schemes are characterized by their nonlinear two-point flux approximation, which is different from the existing ones and has no constraint on the associated interpolation algorithm for auxiliary unknowns. Thanks to the new nonlinear two-point flux formulation, it is no longer required that the interpolation algorithm should be a positivity-preserving one. The first scheme employs vertex unknowns as the auxiliary ones, and a second-order but not positivity-preserving interpolation algorithm is utilized. The second scheme uses the so-called harmonic averaging points located on cell edges to define the auxiliary unknowns, and a second-order positivity-preserving interpolation method is employed. Both schemes have nearly the same convergence rates as compared with their related secondorder linear schemes. Numerical results demonstrate that the new schemes are monotone, and have the second-order accuracy for the solution and first-order for its gradient on severely distorted meshes.