Multifrequency radiation diffusion equations for homogeneous, refractive, lossy media and their interface conditions

We derive time-dependent multifrequency diffusion equations for homogeneous, refractive lossy media. The equations are applicable for a domain composed of several materials with distinct refractive indexes. In such applications, the fundamental radiation variable, the intensity I, is discontinuous across material interfaces. The diffusion equations evolve a variable ξ, the integral of I over all directions divided by the square of the refractive index. Attention is focused on boundary and internal interface conditions for ξ. For numerical solutions using finite elements, it is shown that at material interfaces, the usual diffusion coefficient 1/3κ of the multifrequency equation, where κ is the opacity, is modified by a tensor diffusion term consisting of integrals of the reflectivity. Numerical results are presented. For a single material simulation, the ξ equations yield the same result as diffusion equations that evolve the spectral radiation energy density. A second simulation solves a test problem that models radiation transport in a domain comprised of materials with different refractive indexes. Results qualitatively agree with those previously published.