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 pre-conditioned Generalized Minimum Residual (GMRES) solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for a benchmark problem. 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 4 P3 equations requires 14.5% additional CPU time, while the solution of the single P1 equation requires 9.3% additional CPU time over the 10 equations that are solved for the reacting flow calculations.
Accuracy of compact-stencil interpolation algorithms for unstructured mesh finite volume solver
