Asymptotic Derivation of the Multigroup P₁ and Simplified P_N Equations with Anisotropic Scattering

The multigroup P₁ and simplified P_N (SP_N) equations are derived by an asymptotic expansion of the multigroup transport equation with anisotropic scattering. The P₁ equations are the leading-order approximation in this expansion; the SP_N equations for N = 2,3,… are increasingly higher order approximations. The physical assumptions underlying these approximations are that the material system is optically thick, the probability of absorption is small, and the mean scattering angle is not close to unity. For multigroup isotropic scattering transport problems, a dispersion analysis is given that verifies the accuracy of the SP_N approximations. Numerical comparisons of P₁, SP_N, and S_N solutions are also given. These comparisons show that for low N, SP_N solutions are significantly more accurate (transportlike) than P₁ solutions and are obtained at a significantly lower computational cost than S_N solutions.