A numerical discretization scheme, in both space and time, is considered for the equation of radiative transfer and its corresponding diffusion approximation. Numerical results are presented for radiation penetration into a cold slab driven by a constant incident surface intensity. A comparison of results is made among solutions obtained from the discretization of the radiative transfer equation, a flux-limited diffusion approximation, and the classical diffusion approximation. By numerically studying the properties of the flux-limited diffusion approximation, we conclude that the treatment of the nonlinearities in such a description can significantly affect the results. Different iteration strategies of such nonlinearities are discussed and benchmark data for the converged solution are presented in three different time regimes. Finally, we conclude from this analysis that flux limiting is an important factor in solving these types of problems and must be included in any diffusive description.