  We present a systematic study of the similarity solutions for the Marshak
wave problem, in the local thermodynamic equilibrium (LTE) diffusion
approximation and in the supersonic regime. Self-similar solutions exist for a
temporal power law surface temperature drive and a material model with power
law temperature dependent opacity and energy density. The properties of the
solutions in both linear and nonlinear conduction regimes are studied as a
function of the temporal drive, opacity and energy density exponents. We show
that there exists a range of the temporal exponent for which the total energy
in the system decreases, and the solution has a local maxima. For nonlinear
conduction, we specify the conditions on the opacity and energy density
exponents under which the heat front is linear or even flat, and does posses
its common sharp character; this character is independent of the drive
exponent. We specify the values of the temporal exponents for which analytical
solutions exist and employ the Hammer-Rosen perturbation theory to obtain
highly accurate approximate solutions, which are parameterized using only two
numerically fitted quantities. The solutions are used to construct a set of
benchmarks for supersonic LTE radiative heat transfer, including some with
unusual and interesting properties such as local maxima and non sharp fronts.
The solutions are compared in detail to implicit Monte-Carlo and
discrete-ordinate transport simulations as well gray diffusion simulations,
showing a good agreement, which highlights their usefulness as a verification
test problem for radiative transfer simulations.
