Diffusion Theory with Discontinuities

It has recently been argued that in planar geometry, P₂ theory is more accurate (but no more complex) than P₁ (diffusion) theory as an approximation to transport theory. This argument was based upon analytic comparisons as well as results from numerical test problems. On the analytic side, the P₂ fundamental decay length is more accurate than the corresponding P₁ decay length. One of the purposes of this paper is to show that the P₂ expansion is, in fact, the optimal choice taken from a large family of expansions in predicting this decay length. Further, P₂ theory exhibits scalar flux discontinuities at material interfaces, which can be considered as accounting for internal transport boundary layers. By contrast, the P₁ scalar flux is everywhere continuous. The main purpose of this paper is to present an entire family of diffusion equations that contain flux discontinuities at material interfaces All members of this family predict the exact transport fundamental decay length (the discrete Case eigenvalue). One preferred member of this family is shown to be exceedingly accurate in predicting various transport theory behavior for homogeneous source-free problems. The formalism used to derive these diffusion theories is the variational calculus, including boundary considerations that lead to the diffusive boundary conditions.