The purpose of this thesis is to solve the Ginzburg-Landau equations for the type I and type II superconductors by numerical techniques. We minimize the Gibbs free energy directly, instead of minimizing it analytically by solving the resulting nonlinear partial differential equations. In the lowest free energy state the behavior of the order parameter, the supervelocity, and the internal magnetic field are revealed. We present simulations for a type I superconductor where all function depend only on one coordinate, which agree well with the theory. For the type II superconductors, the simulations are more complicated and we would require a high performance computer system.