A Family of Linearity-Preserving Schemes for Anisotropic Diffusion Problems on General Grids

Abstract

ABSTRACTWe propose a new family of cell-centered finite volume schemes with compact stencils for anisotropic diffusion problems on arbitrary polygonal grids. The derivation of the schemes is based on a general framework through a certain linearity-preserving approach, so that piecewise linear solutions are preserved. These schemes have only one primary cell-centered unknown for each cell, while the so-called harmonic averaging points located at the cell interfaces are employed as auxiliary ones. The main characteristics of these new schemes is the simple and locally defined numerical fluxes that have a common fixed and compact stencil, for example, a nine-point stencil for structured quadrilateral meshes. Moreover, the stability result in H1 norm is obtained theoretically under quite general assumptions, which is most important for the schemes since they are generally asymmetric. Numerical experiments show that the new schemes have nearly second-order accuracy in L2 norm and first-order accuracy in H1 norm on distorted polygonal grids in case that the diffusion tensor is anisotropic and discontinuous.