A numerical study is made of the phenomena of Ostwald ripening in a two-dimensional binary alloy, using the phase-field model employed by Warren and Boettinger. Using this method we are able to trace the history of complicated dynamics driven by surface tension. Coalescence occurs in these systems without any special modification of the phase-field equations, and is observed over a wide range of volume fractions. The results describe an experimental regime which is far removed from the domain of applicability of classical (LSW) coarsening theory. In particular, the volume fraction (area fraction) changes by a large amount over the history of the simulation, resulting in substantial deviations from classical theory.