Electromagnetic wave scattering from planar dielectric films deposited on one-dimensional, randomly rough, perfectly conducting substrates is studied by numerical simulations for both p- and s-polarization. The reduced Rayleigh equation, which is the integral equation satisfied by the scattering amplitude after eliminating the fields inside the film, is the starting point for the simulation. This equation is solved numerically by considering a random surface of finite length, and by introducing wave number cut-offs in the evanescent part of the spectrum. Upon discretization, a system of linear equations is obtained, and by solving this matrix system for an ensemble of surface realizations, the contribution to the mean differential reflection coefficient from the incoherently scattered field, [partial derivative R-nu/partial derivative theta](incoh) (nu = p,s), is obtained nonperturbatively. It is demonstrated that when the scattering geometry supports at least two guided waves, [partial derivative R-nu/partial derivative theta](incoh), has, in addition to the well-known enhanced backscattering peak, well-defined satellite peaks in agreement with theory, for most of the parameters considered.