Variational Derivation of Discrete Ordinate-Like Approximations

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 S_N quadratures do not show significant variations in performance from the parent S_N schemes. In the second portion of the paper, a new “modified” discrete ordinate approximation, MD_N, 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 P_N, and standard P_N approximations. The MD_N results do show a mitigation of the ray effects associated with standard discrete ordinate calculations (D_N), but for gross region-wise absorption rates its accuracy for low orders is more like that of simplified P_N rather than of P_N or D_N approximations. It is concluded that a low-order MD_N approximation should not be a candidate to replace diffusion theory. The approximation may, however, have some application as a calculational standard.