On SN-PN Equivalence

Abstract

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 and 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, and , such that if ψ is the solution of the SN equations with source S+π*(S), then π K ψ is solution of the PN equations with source π K S. We proceed then to prove this Proposition by constructing the two components of the artificial source. We also prove that when the morphism π* 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.