Asymptotic Diffusion and Simplified PN Approximations for Diffusive and Deep Penetration Problems. Part 1: Theory

The classic diffusion approximation to the linear Boltzmann equation is known to be accurate if the underlying physical system is (i) optically thick and (ii) scattering-dominated. This approximation has been mathematically justified by an asymptotic analysis having a scaling that is consistent with these two conditions. Also, the Simplified P N (SP N ) equations have been shown to be higher-order asymptotic corrections to the diffusion equation for the same class of physical problems. In this paper, we alter the asymptotic scaling that yields the standard diffusion and SP N approximations and obtain modified diffusion and SP N approximations that can be significantly more accurate for deep penetration problems, which are not scattering-dominated.