Using simulations, we have studied the coarsening of an ensemble of atom-high two-dimensional islands on a square lattice. The parameters are chosen to represent metal-on-metal homogeneous epitaxy. We study first the elementary processes controlling coarsening: the evaporation of atoms from islands of various sizes and the migration of these islands due to material transport along their edges. The information obtained from these preliminary studies is used to produce a very efficient simulation of coarsening. This allows us to study how coarsening modifies the number of islands, the distribution of their sizes, and the positions of the islands on the surface.