The positivity‐preserving finite volume scheme with fixed stencils for anisotropic diffusion problems on general polyhedral meshes

Abstract

AbstractTwo kinds of nonlinear cell‐centered positivity‐preserving finite volume schemes are proposed for the anisotropic diffusion problems on general three‐dimensional polyhedral meshes. First, the one‐sided flux on the cell‐faces is discretized using the fixed stencil of all vertices, then the cell‐centered discretization scheme is obtained using the nonlinear two‐point flux approximation. On this basis, a new explicit weighted second‐order vertex interpolation algorithm for arbitrary polyhedral meshes is designed to eliminate the vertex auxiliary unknowns in the scheme. In addition, an improved Anderson acceleration algorithm is adopted for nonlinear iteration. Finally, some benchmark examples are given to verify the convergence and positivity‐preserving property of the two schemes.