Variational Derivation of Multigroup P₂ Equations and Boundary Conditions in Planar Geometry

Historically, the even-order P_N equations have been considered a less accurate approximation to the transport equation than the odd-order P_N-1 equations. This perception has stemmed from two apparent conceptual difficulties imposed by the even-order P_N methods— the difficulty in prescribing rigorous boundary conditions for even-order P_N equations that contain the odd number of angular flux moments and the discontinuous character of the even-order P_N solutions at material interfaces. With the first one of the mentioned even-order P_N conceptual problems, a presentation is made of a straightforward and physically-motivated variational procedure based on a new functional that leads from a multigroup planar geometry transport problem to a multigroup P₂ problem with clearly and rigorously defined multigroup boundary conditions. These boundary conditions are new and allow neutron transfer between energy groups at the boundary. These boundary conditions are tested by comparing P₂, P₁, and S_N calculations. Our results show that in the test problems considered, the multigroup P₂ equations with variational boundary conditions are always more accurate than the P₁ multigroup equations with Federighi-Pomraning or Marshak boundary conditions applied to each energy group.