Diffusion process is very important in many coupled systems for real life applications such as radiation hydrodynamics, plasma physics, reservoir modelling and so on. Finite volume discretizations of such systems are very popular among the scientists and engineers due to their nice properties such as local conservation. In many situations, one of the significant features of the finite volume scheme is that the discrete solution should be non-negative. The severely distorted meshes (often occur in the simulation of inertial confinement fusion) and highly anisotropic media (in reservoir modelling) may result in the violation of positivity-preserving property. Many scientists have made great efforts on the construction of positivity-preserving finite volume schemes of diffusion equation that approximately have a second-order accuracy on severely distorted grids in case that the diffusion tensor is taken to be highly anisotropic, at times heterogeneous, and/or discontinuous.

In this talk, two kinds of nonlinear cell-centered positivity-preserving finite volume schemes are proposed for radiation diffusion problems on general three-dimensional polyhedral meshes. Firstly, the one-sided flux on the cell-faces is discretized using the fixed stencil of all vertices. Then by using the nonlinear two-point flux approximation, the cell-centered discretization scheme is obtained by eliminating the vertex auxiliary unknowns. 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 the nonlinear iteration. Finally, some benchmark examples are given to verify the convergence and positivity-preserving property of the two discretization schemes.