Diffusion approximation of radiative transfer problems with interfaces

We derive the diffusion approximation of transport equations with discontinuities at interfaces. The transport equations model the energy density of acoustic waves. The waves are reflected and transmitted at the interface between different media, which leads to discontinuities of the energy density across the interface. The diffusion approximation, which is valid inside each region is not correct at the vicinity of the interface. However, using interface layer analysis, we prove that the transport solution can be approximated by a diffusion term plus an interface layer which decays exponentially fast. We derive systematically the correct form of the interface conditions for this diffusion term.