Modified-Levermore–Pomraning equation: Its derivation and its limitations

Abstract

A transport equation for the mean flux in spatially random media is derived, and is referred to as Modified-Levermore–Pomraning equation (M-L–P). It differs from the conventional L–P equations in that | μ| in the latter is replaced by μ in M-L–P. It is shown that when scattering is present the L–P equations are always incorrect in the sense there is not any special situation in which they can lead to an exact result. In particular they always predict the relaxation lengths of the spatial modes incorrectly. On the other hand, the M-L–P equations are exact when the flux at the origin is deterministic, as in some special cases such as half-infinite medium, and infinite medium with a localized source at the origin, when the density of the medium is spatially random. However, the M-L–P equations become approximate when the medium is a finite slab because of the right boundary condition. But the relaxation rates of the spatial modes are always calculated exactly even in finite slab. The nature of approximation inherent in the M-L–P is elucidated by comparison with the exact “ stochastic transition matrix formalism” developed earlier in two-stream transport.