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 appro ximation to transport theory. This argument was based upon analytic co mparisons as well as results from numerical test problems. On the anal ytic side, the P2 fundamental decay length is more accurate than the c orresponding 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 boundar y layers. By contrast, the P1 scalar flux is everywhere continuous. Th e main purpose of this paper is to present an entire family of diffusi on equations that contain flux discontinuities at material interfaces. All members of this family predict the exact transport fundamental de cay length (the discrete Case eigenvalue). One preferred member of thi s family is shown to be exceedingly accurate in predicting various tra nsport theory behavior for homogeneous source-free problems. The forma lism used to derive these diffusion theories is the variational calcul us, including boundary considerations that lead to the diffusive bound ary conditions.