A variational approximation is developed for general-geometry multigroup transport problems with arbitrary anisotropic scattering. The variational principle is based on a functional that approximates a reaction rate in a subdomain of the system. In principle, approximations that result from this functional ''optimally'' determine such reaction rates. The functional contains an arbitrary parameter alpha and requires the approximate solutions of a forward and an adjoint transport problem. If the basis functions for the forward and adjoint solutions are chosen to be linear functions of the angular variable Omega, the functional yields the familiar multigroup P-1 equations for all values of alpha. However, the boundary conditions that result from the functional depend on alpha. In particular, for problems with vacuum boundaries, one obtains the conventional mixed boundary condition, but with an extrapolation distance that depends continuously on alpha. The choice alpha = 0 yields a generalization of boundary conditions derived earlier by Federighi and Pomraning for a more limited class of problems. The choice alpha = 1 yields a generalization of boundary conditions derived previously by Davis for monoenergetic problems. Other boundary conditions are obtained by choosing different values of alpha. We discuss this indeterminancy of alpha in conjunction with numerical experiments.

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 P-2 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 P-2, P-1, and S-N calculations. Our results show that in the test problems considered, the multigroup P-2 equations with variational boundary conditions are always more accurate than the P-1 multigroup equations with Federighi-Pomraning or Marshak boundary conditions applied to each energy group.

Even-order P(N) theory has historically been viewed as a questionable approximation to transport theory. The main reason is that one obtains an odd number of unknowns and equations; this causes an ambiguity in the prescription of boundary conditions. We derive the one-group planar-geometry P2 equations and associated boundary conditions using a simple, physically motivated variational principle. We also present numerical results comparing P2, P1, and SN calculations. These results demonstrate that for most problems, the P2 equations with variational boundary conditions are considerably more accurate than the P1 equations with either the Marshak or the Federighi-Pomraning boundary conditions (both of which have also been derived variationally). Moreover, because the P2 and P1 equations can be written in diffusion form, the discretized P2 equations require nearly the same computational effort to solve as the discretized P1 equations. Our variational method can easily be extended to higher even-order P(N) approximations.

The P(N) theory has been shown to be an asymptotic limit of transport theory for problems in optically thick planar-geometry media with low absorption rates and highly anisotropic scattering. Transport problems that lie outside the asymptotic regime of validity of P(N) theory are considered. Such problems occur in media that are either optically thin, or contain isotropic or mildly anisotropic scattering, or are not weakly absorbing. For such problems, the accuracy of numerical solutions of the P(N) equations obtained using the asymptotic boundary conditions is demonstrated. These numerical solutions are compared with others obtained using various familiar boundary conditions. Solutions obtained using the asymptotic boundary conditions are always competitive with, and often superior to, solutions obtained using these other boundary conditions.