Few-Group Approximations Based on a Variational Principle

The equations and boundary conditions that constitute the P₁ approximation to the space-time-energy transport equation and its adjoint can be obtained from a variational expression that admits trial functions discontinuous in space and energy. This expression can then be used to derive all the standard forms of the few-group diffusion equations—equations using flux averaged constants, over-lapping group equations, parallel group equations—as well as many more hitherto unexamined. Such a procedure is carried out in the present paper. All the standard few-group results, as well as formally exact few-group equations and multigroup equations, are shown to be special cases of a single general form derived from the variational expression. Internal boundary conditions are obtained automatically, and it is shown that in some cases discontinuities in fluxes and currents ought to be imposed across internal boundaries.