An extensive computer simulation investigation of the time dependence of spinodal decomposition in a two-dimensional, one-component fluid is reported. This investigation includes constant-temperature and constant-energy simulations, both of which are followed to very long times. We observe the detailed density morphology, finding different growth regions for the average size of the formed liquid clusters. The late-time growth law for the average cluster size is found to be t12 for the isothermal and t13 for the constant-energy simulation, respectively. The physical origin of these results is explained by asymptotic analysis of the governing equations of cluster growth. A linear hydrodynamic theory for the early stages of the separation process is also presented. © 1983 The American Physical Society.