The time-dependent Ginzburg-Landau equations (TDGL) model a thin-film superconductor of finite size placed under a magnetic field. For numerical computation, we use a staggered grid discretization, a technique well known in numerical fluid mechanics. Some properties of the solutions are established. An efficient explicit-implicit method based on the forward Euler method is developed. In our simulations, we impose natural boundary conditions at the edge of the superconductor. With suitable choices of parameters (corresponding to physical superconductors of type II) and the strength of the external magnetic field, the steady-state solutions exhibit vortices. When a variable strength magnetic field, simulating a transient current, is introduced, we observe motion of the vortices in a periodic pattern.