This paper studies the numerical solution for anisotropic diffusion equations. In multi-material radiative hydrodynamics problems, the diffusion coefficient is usually a discontinuous and anisotropic tensor. In this paper, an efficient and robust finite volume scheme for diffusion equations with arbitrary anisotropic diffusion coefficients is constructed on arbitrarily polyhedral meshes. The cell-centered scheme is widely used in multi-material radiative hydrodynamics. Based on the idea of linearity-preserving, a class of second-order accurate cell-centered finite volume schemes is derived in this paper, exploring a new approach to constructing three-dimensional diffusion schemes. We first extend the two-dimensional nine-point scheme to a three-dimensional diamond scheme based on the linearity-preserving criterion. The name of the diamond scheme comes from the diamond-shaped stencil used in the flux approximation, and the normal flux on the mesh face is expressed as a combination of the cell unknowns on both sides of the face and the vertex unknowns on the face. Then, we treat the vertex unknowns as auxiliary unknowns and eliminate them by interpolating from the surrounding cell-centered unknowns. We use the linearity-preserving criterion to construct two linearity-preserving vertex interpolation algorithms on three-dimensional polyhedral meshes. The first vertex interpolation algorithm is obtained by combining multi-point flux approximation with a limiting process, and the second vertex interpolation algorithm is obtained by combining least squares techniques with graph search algorithms. Both vertex interpolation algorithms maintain linearity-preserving on arbitrary polyhedral meshes and for arbitrary anisotropic diffusion tensors.
In this paper, we investigate the application of the cell-centered finite volume scheme for the anisotropic diffusion equation in the two-phase flow model in porous media. The mathematical model comprises a diffusion equation for the pressure and a nonlinear hyperbolic equation for the saturation. Here, the diffusion equation for the pressure is a second-order elliptic equation with an anisotropic and eventually discontinuous diffusion coefficient. We develop efficient and robust finite volume schemes for diffusion equations with arbitrary anisotropic coefficients on arbitrary unstructured grids. In this paper, based on the idea of linearity-preserving, a family of second-order cell-centered finite volume scheme will be derived and applied to the numerical simulation of two-phase flow in porous media. The saturation equation is solved using the second-order monotone MUSCL schemes which were widely used in hyperbolic conservation law, and then a two-phase flow simulator with extensible scheme is formed. In addition, the research results of this paper can be applied to the engineering problems of two-phase displacement such as metal casting.
