The first-order spherical harmonics method (or P1 approximation) has found prolific usage for approximate solution of the radiative transfer equation (RTE) in participating media. However, the accuracy of the P1 approximation deteriorates as the optical thickness of the medium is decreased. The Modified Differential Approximation (MDA) was originally proposed to remove the shortcomings of the P1 approximation in optically thin situations. This article presents algorithms to apply the MDA to arbitrary geometry—in particular, three-dimensional (3D) geometry with obstructions, and inhomogeneous media. The wall-emitted component of the intensity was computed using a combined view-factor and ray-tracing approach. The Helmholtz equation, arising out of the medium-emitted component, was solved using an unstructured finite-volume procedure. The general procedure was validated against benchmark Monte Carlo results. The accuracy of MDA was found to be far superior to the standard P1 approximation in optically thin situations, and comparable to the P1 approximation in optically thick situations. Calculation and storage of the view-factor matrix was found to be a major shortcoming of the MDA, and several strategies to reduce both memory and computational time are discussed and demonstrated. In addition, for inhomogeneous media, calculation of optical distances requires a ray-tracing procedure, which was found to be a bottleneck from a computational efficiency standpoint.
Adaptive source biasing sampling for time-dependent radiation transport problems