A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface

When a linear transport equation contains two scales, one diffusive and the other non-diffusive, it is natural to use a domain decomposition method which couples the transport equation with a diffusion equation with an interface condition. One such method was introduced by Golse, Jin and Levermore in [11], where an interface condition, which is derived from the conservation of energy density, was used to construct an efficient non-iterative domain decomposition method.
    In this paper, we extend this domain decomposition method to diffusive interfaces where the energy flux is conserved. Such problems arise in high frequency waves in random media. New operators corresponding to transmission and reflections at the interfaces are derived and then used in the interface conditions. With these new operators we are able to construct both first and second order (in terms of the mean free path) non-iterative domain decomposition methods of the type by Golse-Jin-Levermore, which will be proved having the desired accuracy and tested numerically.