Transient melting is an ubiquitous phenomenon in nature, which plays an increasingly important role in the processing of nanomaterials. A sound theoretical description of this process is therefore important, both from fundamental and applied points of view. We present a numerical study of transient melting in simple atomic solids using both, continuum theory based on the heat diffusion equation and transient nonequilibrium molecular dynamics simulations. We show that continuum theory provides an accurate description of relevant properties, temperature relaxation, time-dependent internal energy, and dynamics of the melting front. However, deviations between the continuum approach and the molecular dynamics simulations are observed in picosecond time scales depending on the initial temperature used to melt the solid. These deviations are due to the emergence of new time scales associated with the activated character of the melting process. Consistently with this notion, we observe that the closer the initial temperature to the melting temperature of the solid, the longer the time it takes for the system to converge to the continuum solution. For systems investigated here we find a delay in the recovery of the continuum solution of ≲5 to ≲80 ps for initial temperatures between 40 and 25% above the melting temperature of the solid, respectively. We find that the combination of continuum theory and molecular dynamics simulations provides a useful approach to quantify the temperature relaxation and the melting temperature of materials using short molecular dynamics trajectories.