A one-dimensional model for simulating localized corrosion is developed, incorporating time-dependent diffusion-migration transport. The model is applied to iron with the Butler-Volmer formula as the dissolution law, and both crevice and pit configurations are simulated. A finite-difference ALE scheme is used for the numerical computation of the solutions of this free boundary problem. The results show that the dissolution rate increases with chloride concentration and metal potential, and migration plays, most of the time, a significant role in species transport for both crevice and pit configurations. The evolution of the repassivation potential with pit depth is computed and is in good agreement with experimental results. The time-dependent model under consideration provides more accurate quantitative results for concentration profile and crevice depth than stationary models.