We present here an efficient numerical scheme for solving the non-relativistic 1D radiation-hydrodynamics equations including inelastic Compton scattering, which is not included in most codes and is crucial for solving problems such as shock breakout
. The devised code is applied to the problems of a steady-state planar radiation mediated shock (RMS) and RMS breakout from a stellar envelope. The results are in agreement with those of a previous work on shock breakout citep{Sapir13}, in which Compton equilibrium between matter and radiation was assumed and the effective photon approximation was used to describe the radiation spectrum. In particular, we show that the luminosity and its temporal dependence, the peak temperature at breakout, and the universal shape of the spectral fluence derived in this earlier work are all accurate. Although there is a discrepancy between the spectral calculations and the effective photon approximation due to the inaccuracy of the effective photon approximation estimate of the effective photon production rate, which grows with lower densities and higher velocities, the difference in peak temperature reaches only $30%$ for the most discrepant cases of fast shocks in blue supergiants. The incompatibility of the stellar envelope shock breakout model results with observed properties of X-ray flashes and the discrepancy between the predicted and observed rates of X-ray flashes citep{Sapir13} remain unexplained.
