The "island dynamics" method was applied to simulation of kinetically limited metal nucleation and growth by electrodeposition in the presence of additives. The model includes additive kinetics (described by a set of ordinary differential rate equations), surface diffusion of adatoms (calculated with a continuum equation), and nucleation (calculated by a rate equation that depends on average adatom concentration and diffusivity). Nuclei are placed on the surface stochastically at locations weighted by the local value of adatom concentration. The moving interface is tracked by the level-set method. The model was demonstrated for copper deposition in acid-sulfate electrolyte containing [bis(3-sulfopropyl)disulfide], polyethylene glycol, and chloride. Numerical results were obtained for fractional coverage of additive species and reaction intermediates, coverage of metal deposit on the surface, and spatial information on nuclei, islands, and multilayer structures. Simulation results were compared with kinetic Monte Carlo (KMC) calculations and found to be within 1% for fractional coverage values and within 10% for nucleation density. The computational speed was 10-30x higher than comparable KMC simulations over the range studied. The accuracy and computational speed of the island dynamics algorithm captures phenomena present at widely varying length and time scales which are needed for molecular engineering of electrodeposition processes.