Variational Derivation of Discrete Ordinate-Like Approximations

Abstract

The derivation of discrete ordinate and discrete ordinate-like approximations from variational principles for the one-speed transport equation is explored here. Standard discrete ordinate approximations are derived from a first-order stationary variational principle. The derivation yields a prescription for ordinates to be used given a selection of weights. Resultant quadrature schemes are compared numerically with those in common use. These new schemes derived using the weights of SN quadratures do not show significant variations in performance from the parent SN schemes. In the second portion of the paper, a new “modified” discrete ordinate approximation, MDN, is found by applying the same techniques as in the derivation of the standard approximation, this time, however, using an extremum second-order variational principle. The new approximation is compared through several numerical examples with standard discrete ordinate, simplified PN, and standard PN approximations. The MDN results do show a mitigation of the ray effects associated with standard discrete ordinate calculations (DN), but for gross region-wise absorption rates its accuracy for low orders is more like that of simplified PN rather than of PN or DN approximations. It is concluded that a low-order MDN approximation should not be a candidate to replace diffusion theory. The approximation may, however, have some application as a calculational standard.