On SN-PN Equivalence

We consider the artificial conversion of the discrete-ordinates (SN) equations into a system of spherical harmonic (PN) equations. This is done by adding to the SN equations an artificial source that has two components. The first component transforms the SN scattering term into PN-like scattering, while the second modifies the SN streaming operator into a lower-order PN streaming operator. Denoting by F and E the spaces of solutions of the SN and PN equations, respectively, we define SN-PN equivalence via a constructive Proposition based on two linear morphisms, pi(K) : F -> E and pi(*) : F -> F, such that if psi is the solution of the SN equations with source S + pi(*)(S), then pi K psi is solution of the PN equations with source pi S-K. We proceed then to prove this Proposition by constructing the two components of the artificial source. We also prove that when dim E < dim F the morphism pi(*) is not unique, and propose a general form for the second component of the artificial source, which is shown to comprise all artificial sources previously proposed in the literature.