Finite-Volume Formulation and Solution of the $P3$ Equations of Radiative Transfer on Unstructured Meshes

The method of spherical harmonics (or $PN$⁠) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the $P3$ equations of radiative transfer in two-dimensional geometry—a set of four coupled, second-order partial differential equations. The FV procedure presented here is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the binary spatial partitioning algorithm, followed by fully implicit coupled solution within each cell group using a preconditioned generalized minimum residual solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for two benchmark problems. For completeness, results using the $P1$ approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and $P3$ results increase as the optical thickness is decreased. The $P3$ approximation is found to be more accurate than the $P1$ approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the four $P3$ equations requires 14.5% additional CPU time, while the solution of the single $P1$ equation requires 9.3% additional CPU time over the case without radiation.