We consider a time-dependent, energy-dependent, nonlinear radiative transfer problem in which opacities are large [O(ฮต -1 )] and interior sources are small [O(ฮต)]. An asymptotic analysis of this problem as ฮต โ 0 leads to the equilibrium diffusion equation in the interior of the system, along with boundary conditions and initial conditions for this equation. We apply the same asymptotic analysis to a discrete version of the problem, in which the frequency variable is discretized by the multigroup method, the direction variable by the discrete-ordinates method, the time variable by the fully implicit method, and the spatial variable by a subcell-balance method. We find that as ฮต โ 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation, with boundary conditions and initial conditions that are remarkably accurate. The analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients, then the numerical method obtains an accurate solution in the interior of the system, even though the optical thickness of the spatial cells tends to โ and boundary layers in the transport solution are not resolved. We go a step further to analyze problems that are optically thin at some photon frequencies but thick at others, and show that once again the discrete solution is remarkably accurate. We present numerical results that verify these and other predictions of the analyses.