This paper studies the numerical solution of diffusion equation in multi-material radiation hydrodynamics. Diffusion coefficients of multi-material problems are usually discontinuous and anisotropic tensors. In this paper, we will construct an efficient and robust finite volume scheme for diffusion equations with arbitrary anisotropic diffusion coefficients on arbitrary distorted polyhedral meshes. The cell-centered finite volume scheme is widely used in multi-material radiation fluid. In this paper, a class of cell-centered finite volume schemes with second-order accuracy will be derived based on the idea of linearity-preserving, and a new way to construct three-dimensional diffusion scheme will be explored. Firstly, the idea of linearity-preserving is applied to construct the vertex interpolation algorithm on 3D polyhedral meshes, which will be the first second-order accurate vertex interpolation algorithm for 3D large deformation meshes. Then, based on the vertex interpolation algorithm, the 2D nine point scheme is extended to 3D.
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.
