A multidimensional unified gas-kinetic scheme for radiative transfer equations on unstructured mesh

In order to extend the unified gas kinetic scheme (UGKS) to solve radiative transfer equations in a complex geometry, a multidimensional asymptotic preserving implicit method on unstructured mesh is constructed in this paper. With an implicit formulation, the CFL condition for the determination of the time step in UGKS can be much relaxed, and a large time step is used in simulations. Differently from previous direction-by-direction UGKS on orthogonal structured mesh, on unstructured mesh the interface flux transport takes into account multi-dimensional effect, where gradients of radiation intensity and material temperature in both normal and tangential directions of a cell interface are included in the flux evaluation. The multiple scale nature makes the UGKS be able to capture the solutions in both optically thin and thick regions seamlessly. In the optically thick region the condition of cell size being less than photon's mean free path is fully removed, and the UGKS recovers a solver for diffusion equation in such a limit on unstructured mesh. For a distorted quadrilateral mesh, the UGKS goes to a nine-point scheme for the diffusion equation, and it naturally reduces to the standard five-point scheme for a orthogonal quadrilateral mesh. Numerical computations covering a wide range of transport regimes on unstructured and distorted quadrilateral meshes will be presented to validate the current approach.