Computation of Interface Reflection and Regular or Diffuse Transmission of the Planar Symmetric Radiative Transfer Equation with Isotropic Scattering and Its Diffusion Limit

In this paper, we develop a numerical scheme for the interface problem in the planar symmetric radiative transfer equation with isotropic scattering. Such problems arise in the modeling of the propagation of energy density for waves in heterogeneous media with weak random fluctuation in the high frequency regime. The idea, following the earlier work of Jin and Wen for regular transmission and reflection, is to build the interface condition, which characterizes the reflection and transmission, into the numerical flux. The new contribution of this article is to deal with the diffuse transmission at the interface. The positivity of the scheme is proven. Moreover, we show, via numerical examples, that this new scheme is able to capture the correct (regular or diffuse) transmission and reflection through the interface, and, as the mean free path goes to zero, captures the diffusion limit with the correct interface conditions.