A consistent approach to solving the radiation diffusion equation

Diffusive x-ray-driven heat waves are found in a variety of astrophysical and laboratory settings, e.g., in the heating of a hohlraum used for inertial confinement fusion, and hence are of intrinsic interest. However, accurate analytic diffusion wave (also called Marshak wave) solutions are difficult to obtain due to the strong nonlinearity of the radiation diffusion equation. The typical approach is to solve near the heat front, and by ansatz apply the solution globally. This approach works fairly well due to “steepness” of the heat front, but energy is not conserved and it does not lead to a consistent way of correcting the solution or estimating accuracy. In this work, the steepness of the front is employed through a perturbation expansion in e=β/(4+α), where the internal energy varies as Tβ and the opacity varies as T−α. The equations are solved using an iterative approach, equivalent to asymptotic methods that match outer (away from the front) and inner (near the front) solutions. Typically e<0.3. Ca...