Simulations are reported to investigate solid superheating and liquid supercooling of two-dimensional systems with a Yukawa interparticle potential. Motivated by experiments where a dusty plasma is heated and then cooled suddenly, we track particle motion using a simulation with Langevin dynamics. Hysteresis is observed when the temperature is varied rapidly in a heating and cooling cycle. As in the experiment, transient solid superheating, but not liquid supercooling, is observed. Solid superheating, which is characterized by solid structure above the melting point, is found to be promoted by a higher rate of temperature increase.