Further Development of the Elliptic PDE Formulation of the P_N Approximation and its Marshak Boundary Conditions

Abstract

The expansion of the radiative transfer equation (RTE) into spherical harmonics results in the P _N approximation, consisting of (N + 1)² simultaneous, first-order partial differential equations (PDEs). This system of equations is generally solved subject to a set of so-called Marshak's boundary conditions, although some ambiguity exists in multidimensional media, for which the set provides more than the necessary number of conditions. In recent work Modest has shown that the general 3-D P _N approximation can be formulated as a set of N(N + 1)/2 second-order, elliptic PDEs, using the original set of Marshak's conditions, and which can be solved with standard PDE solution packages. In this article the Marshak boundary conditions are reexamined in the light of the elliptic formulation, culminating in a self-consistent set of N(N + 1)/2 conditions along the boundary of the enclosure. The elliptic set of PDEs is reformulated and reduced considerably by limiting considerations to isotropic scattering. As an example, the 2-D P ₃ approximation is extracted, and sample 2-D P ₁, P ₃, and P ₅ computations are compared with Monte Carlo results.

Acknowledgments

The majority of the work presented here was carried out by the author while at the University of Karlsruhe, Germany, in the framework of a Humboldt Research Award. The author acknowledges the support provided by both the University of Karlsruhe and the Alexander von Humboldt Foundation. In addition, Dr. S. Lei is acknowledged for providing the Monte Carlo results.

Notes

¹Note that, for convenience, the notation for used here differs by a factor of 4π from that used in [8].

²In Blanco's derivation, a normalization factor is employed. In order to be consistent with the real spherical harmonics used in the current study, a modification coefficient was included in the transformation.