A numerical approach for non-linear acoustic wave propagation is proposed in which a finite element method is used. An approximate equation which can describe the effect of diffraction with reasonable accuracy is derived based on the equations of fluid dynamics. Only the generation of the second harmonic wave is considered, with the higher order harmonics being neglected. Under the assumption of weak non-linearity, a set of uncoupled equations for the primary and secondary wave is discretized in space by a finite element method, and then solved by using the Newmark-β integration scheme for time. Only two-dimensional cases are considered, and some numerical examples are presented for sound propagation along a duct with stepped width, for scattering from a circular rod and for a focusing source in which the diffraction effect is of interest. Some examples of the wave propagation fields are displayed.