The PN and double PN approximations to the monoenergetic transport equation are modified to improve their asymptotic behaviour. Free parameters are introduced into the PN and double PN equations through the use of a pseudo-scattering kernel. The parameters are evaluated by stipulating that the resulting scalar flux approximations to the infinite-medium Green's function yield exact asymptotic solutions while conserving specified spatial moments. Asymptotic P3 and double P1 solutions are presented for the infinite-medium and half-space Green's functions and for the source-free and constant-source Milne problems.
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