The two-region Milne problem

We consider the two-region Milne problem, defined as the steady-state monoenergetic linear transport problem for two adjoining homogeneous source-free half-spaces, with a particle source coming from infinity in one of the half-spaces. We demonstrate that the asymptotic (Case discrete mode) component of the solution for the scalar flux is easily and explicitly written in terms of Chandrasekhar's H-function for each medium. This asymptotic solution is shown to exhibit a discontinuity in both the scalar flux and current at the interface between the two half-spaces. Numerical benchmark results for the linear extrapolation distance and the discontinuities are given for various combinations of the mean number of secondaries (c) characterizing the two media. Contact is also made with a variational treatment. In particular, the variational formalism is shown to predict the linear extrapolation distance and these asymptotic discontinuities correct to first order in the difference between the values of c characterizing the two half-spaces.