  A higher order Godunov method for the radiation subsystem of radiation
hydrodynamics is presented. A key ingredient of the method is the direct
coupling of stiff source term effects to the hyperbolic structure of the system
of conservation laws; it is composed of a predictor step that is based on
Duhamel's principle and a corrector step that is based on Picard iteration. The
method is second order accurate in both time and space, unsplit, asymptotically
preserving, and uniformly well behaved from the photon free streaming
(hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and
to the strong equilibrium diffusion (hyperbolic) limit. Numerical tests
demonstrate second order convergence across various parameter regimes.
