Hybrid S_N-P_N Solution of the Radiative Transfer Equation in Multi-Dimensional Media

The Modified Differential Approximation (MDA) was originally proposed for solution of the radiative transfer equation (RTE) in order to remove the shortcomings of the P₁ approximation in scenarios where the radiation intensity is strongly directionally dependent. In the original MDA approach, the wall-emitted component of the intensity is determined using a surface-to-surface exchange formulation that makes use of geometric viewfactors. Such an approach is computationally very expensive for complex geometry and/or inhomogeneous media. This article presents a new formulation in which the wall-emitted component is solved using the Discrete Ordinates Method (S_N approximation), while the medium-emitted component is solved using the P₁ approximation, resulting in a hybrid S_N-P_N RTE solver. Results show that the hybrid Discrete Ordinates-P₁ method (DOM-P₁) is computationally very efficient, but its accuracy is poor in optically thin situations where ray effects, inherent in the Discrete Ordinates Method, are pronounced. To circumvent this problem, the control-angle Discrete Ordinates Method (CADOM) is finally employed, and the accuracy of the hybrid CADOM-P₁ method is found to be far superior to the hybrid DOM-P₁ method.