On SPN theory. (July 2019) - Sanchez, Richard
Highlights: Review of SPN theoretical literature showing progress in understanding of SPN method. Generalization of Pomraning's variational analysis of SPN equations. New SPN paradigm based on a set of coupled integral equations for N + 1 scalar fluxes. New equations provide an expression for the SPN angular flux as subset of PN flux. Interface conditions assure regularity of SPN solution through the interface. Abstract: We have generalized Pomraning's variational derivation of the Simplified P N (SP N ) method and apply it to the first-order transport equation with anisotropic scattering and sources without restriction on the parity of N . The result is a new system of coupled linear integral equations for N + 1 scalar functions, which are shown to be equivalent to the traditional first-order and second-order (even or odd) SP N equations. The integral operator in these equations is the square root of the Laplacian. We view these integral equations as a natural paradigm for SP N in that they do not introduce the artificial current vectors which add two artificial degrees of freedom (per current) to the traditional first-order SP N equations. Interface conditions are derived by requiring the solution to behave smoothly across the interface between homogeneous regions. Surprisingly, these conditions are identical to the traditional ones obtained assuming a slab-like behavior for the angular flux. In an infinite homogeneous medium the integral SP N equations give the exact P N solution and here we join the work of Ackroyd et al. and Chao. We have also derived a simple recurrence relation to obtain the well-known PDE for the scalar flux. By virtue of the equivalence SP N -P N in an infinite homogeneous medium, this technique gives the scalar-flux PDE for P N, which was obtained for N = 1, 2, 3 by Davison in a much more involved way. We include also a comparative review of most of the derivations of the SP N equations in the literature, including the recent work by Chao, which shows the progress in the understanding of these equations which culminates with Chao's work. Our integral equations are fully equivalent to those of Chao, in that they predict the same equations and associated SP N angular flux.