Advanced Differential Approximation Formulation of the P_N Method for Radiative Transfer

The spherical harmonics (P_N) method, especially its lowest order, i.e., the P₁ or differential approximation, enjoys great popularity because of its relative simplicity and compatibility with standard models for the solution of the (overall) energy equation. Low-order P_N approximations perform poorly in the presence of strongly nonisotropic intensity distributions, especially in optically thin situations within nonisothermal enclosures (due to variation in surface radiosities across the enclosure surface, causing rapid change of irradiation over incoming directions). A previous modification of the P_N approximation, i.e., the modified differential approximation (MDA), separates wall emission from medium emission to reduce the nonisotropy of intensity. Although successful, the major drawback of this method is that the intensity at the walls is set to zero into outward directions, while incoming intensity is nonzero, resulting in a discontinuity at grazing angles. To alleviate this problem, a new approach, termed here the “advanced differential approximation (ADA),” is developed, in which the directional gradient of the intensity at the wall is minimized. This makes the intensity distribution continuous for the P₁ method and mostly continuous for higher-order P_N methods. The new method is tested for a 1-D slab and concentric spheres, and for a 2-D medium. Results are compared with the exact analytical solutions for the 1-D slab as well as the Monte Carlo-based simulations for 2-D media.