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 epsilon=beta/(4+alpha), where the internal energy varies as T-beta and the opacity varies as T-alpha. 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 epsilon<0.3. Calculations through first order in epsilon and are accurate to similar to10%, which is comparable to the inaccuracy from assuming power laws for material properties. Supersonic waves with arbitrary drive time history are solved for, including the case of a rapidly cooling surface. The method is then generalized to arbitrary temperature dependence of opacity and internal energy. Also solved for are subsonic waves with drive temperature varying as a power of time. In the subsonic case, the specific heat (pressure/density) and opacity are each assumed to vary as density to a small power, of order epsilon. Solutions are obtained through order epsilon(2) and it is found that the theory compares well with radiation hydrodynamics code calculations of the heat front position, absorbed flux, and ablation pressure. (C) 2003 American Institute of Physics.