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The multigroup P1 and simplified PN (SPN) 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 SPN 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 SPN approximations. Numerical comparisons of P1, SPN, and SN solutions are also given. These comparisons show that for low N, SPN solutions are significantly more accurate (transportlike) than P1 solutions and are obtained at a significantly lower computational cost than SN solutions.
Larsen et al. (Mon,) studied this question.