The Godunov method that tracks nonstationary fronts and interfaces as boundaries of subregions moving with time is extended to include radiation transport. In each subregion and at each time step, a new grid is created by use of boundary-fitted coordinates. The radiation transport is performed in a multiangle-multifrequency approach. The numerical method is based on a finite volume method in the space time-domain, and the hydrodynamic fluxes are calculated using the solution of Riemann problems. Numerical results are shown for some selected problems to demonstrate the efficiency of this approach.