P₁, P₂ and Asymptotic Approximations for Stochastic Transport

The problem of describing particle transport through a Markovian stochastic mixture of two immiscible materials is generally approximated by the so-called Levermore model, consisting of two coupled transport equations. In this paper, the P₂ diffusive equations and the associated boundary conditions for this Levermore model are derived in planar geometry by using a variational principle, and numerical results comparing P₂, P₁, and S₁₆ (benchmark) calculations are presented. These results demonstrate that the P₂ equations are considerably more accurate than the P₁ equations away from boundary layers. An asymptotic diffusion approximation to this model is also explored with several different boundary conditions, and the overall conclusion is that the asymptotic diffusion treatment is in general inferior to P₂ theory, and its superiority over P₁ theory is not overwhelming and not consistent.