We have developed a two-dimensional numerical model of excimer-laser melting and solidification that properly takes into account the non-equilibrium and transient nature of the process. The model incorporates a novel explicit finite difference scheme for efficiently solving the heat conduction equation and an algorithm that incorporates the interface response function for properly simulating the evolution of phase domains. The model provides space- and time-resolved information regarding the thermal profile and phase domains from which nearly all of the important solidification details can be extracted (e.g., interface location, solidification velocity, interfacial undercooling, etc.). For the simple partial-melting-and-vertical-regrowth scenario, results from the model converge with the results from the well-established one-dimensional model. As a result of its two-dimensional and non-equilibrium formulation, which also respects the amorphous and inert nature of the underlying oxide surface, the model is unique in its capability for properly simulating those solidification scenarios that involve extensive lateral growth of solids, as for example those behind the super-lateral growth phenomenon and various artificially controlled super-lateral growth processes.