The VEM-based positivity-preserving conservative scheme for radiation diffusion problems on generalized polyhedral meshes

Abstract

In this paper, for the 3D radiation diffusion equation on generalized polyhedral meshes, we construct a positivity-preserving conservative scheme based on the virtual element method (VEM). Firstly, the equation is discretized by finite volume, and the one-sided flux on the cell-face is discretized by using the fixed stencil of all vertices. Then, the nonlinear two-point flux approximation is used to obtain the equations about cell-centered unknowns and vertex unknowns; the lowest order VEM solves the vertex values of the mesh. Finally, the conservative equations containing only cell-centered unknowns are solved. This scheme ensures the high accuracy of VEM and embodies the local conservation and positivity-preserving. Compared with the existing nonlinear positivity-preserving finite volume schemes, the new scheme does not need nonlinear iteration for linear problems. For nonlinear problems, it only needs to use the VEM to solve the nonlinear system. Numerical results show that the new scheme can achieve the optimal convergence order and maintain the nonnegativity of the solution for generalized polyhedral meshes and arbitrary diffusion tensors.