For modelling the propagation of electromagnetic waves in the low troposphere, the discrete mixed Fourier method is classically used. It is based on a finite-difference approximation of the ground boundary condition. However, the propagator of the split-step Fourier method is derived from continuous equations. In this paper, we apply the finite-difference approximation to the propagation equation. A discrete operator is derived from these discrete equations. The continuous and discrete propagators are compared. The accuracy of these propagators is numerically analysed in free space and over an impedance ground. We show that the discrete propagator avoids some numerical instabilities.