Nonlinear (large amplitude) vibrations of thin elastic plates can exhibit strongly nonlinear regimes characterized by a broadband Fourier spectrum and a cascade of energy from the large to the small wavelengths. This particular regime can be properly described within the framework of wave turbulence theory. The dynamics of the local kinetic energy spectrum is here investigated numerically with a finite difference, energy-conserving scheme, for a simply-supported rectangular plate excited pointwise and harmonically. Damping is not considered so that energy is left free to cascade until the highest simulated frequency is reached. The framework of non-stationary wave turbulence is thus appropriate to study quantitatively the numerical results. In particular, numerical simulations show the presence of a front propagating to high frequencies, leaving a steady spectrum in its wake, which has the property of being self-similar. When a finite amount of energy is given at initial state to the plate which is then left free to vibrate, the spectra are found to be in perfect accordance with the log-correction theoretically predicted. When forced vibrations are considered so that energy is continuously fed into the plate, a slightly steeper slope is observed in the low-frequency range of the spectrum. It is concluded that the pointwise forcing introduces an anisotropy that have an influence on the slope of the power spectrum, hence explaining one of the discrepancies reported in experimental studies.