Asymptotically consistent reflection boundary conditions for diffusion theory

It is well known that the classic diffusion equation is an asymptotic limit of the one-speed linear transport equation. In this paper, we carry out the analysis required to obtain an asymptotically consistent boundary condition for this diffusion equation in the case of partial surface reflection. The specification of this boundary condition requires the transport theory solution to a purely scattering halfspace problem. A variational treatment is used to obtain explicit results for this halfspace problem in the general case. In the special case of purely diffuse reflection, simple considerations give exact halfspace results, and a more involved exact halfspace analysis is presented in the case of specular reflection. These exact results are used to assess the accuracy of the variational treatment.