On SPN theory

Abstract

We have generalized Pomraning’s variational derivation of the Simplified PN (SPN) 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) SPN equations. The integral operator in these equations is the square root of the Laplacian. We view these integral equations as a natural paradigm for SPN in that they do not introduce the artificial current vectors which add two artificial degrees of freedom (per current) to the traditional first-order SPN 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 SPN equations give the exact PN 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 SPN-PN in an infinite homogeneous medium, this technique gives the scalar-flux PDE for PN, 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 SPN 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 SPN angular flux.