The P_N Theory as an Asymptotic Limit of Transport Theory in Planar Geometry —I: Analysis

The P_N theory is shown to be an asymptotic limit of transport theory for an optically thick planar-geometry system with small absorption and highly anisotropic scattering. The asymptotic analysis shows that the solution in the interior of the system is described by the standard P_N equations for which initial, boundary, and interface conditions are determined by asymptotic initial, boundary layer, and interface layer calculations. The asymptotic initial, (reflecting) boundary, and interface conditions for the P_N equations agree with conventional formulations. However, at a boundary having a prescribed incident flux, the asymptotic boundary layer analysis yields P_N boundary conditions that differ from previous formulations. Numerical transport and P_N results are presented to substantiate this asymptotic theory.