Transient and steady-state amperometric tip responses were simulated with an alternating direction implicit algorithm. Compared with previous publications, the simulation domain was designed to account for the diffusion of the redox species around the corner of the insulating sheath. Expanding space and time grids were used to optimize the algorithm, and the simulation was validated by comparison with published data for the microdisk electrode. Tip responses were simulated for a wide range of tip substrate distances over conducting and insulating substrates. The shape of the approach curves was investigated for several electroactive disk to insulator radii ratios. Diffusion around the edge of the insulating sheath was found to have a pronounced effect on the approach curves. In contrast to the findings of earlier studies, tip currents for conducting substrates were found to significantly depend on the tip geometry. The parameters of functions used to describe approach curves in the SECM literature were studied for several tip geometries commonly used experimentally. Simulated results were also used to assess the topographical sensitivity (the rate of change of tip current with respect to tip-substrate distance) and spatial resolution (the ability of the microdisk to distinguish two conducting islands inlaid into an insulating substrate) of the scanning electrochemical microscope (SECM).