Asymptotically exact boundary conditions for the P_N equations

It was recently demonstrated that in planar geometry, the classic P_N equations are an asymptotic limit of the transport equation. A corresponding boundary layer analysis established the asymptotically consistent boundary conditions. These boundary conditions were evaluated variationally, and it was conjectured that these variational approximations are quite accurate for all values of N. Here, we evaluate these boundary conditions exactly (numerically) and show that the previous variational results are indeed accurate to a few percent. The exact results were computed using numerical methods previously developed for solving Chandrasekhar's H equations.