The time dependent Ginzburg-Landau (TDGL) equation is a typical model in phase field theory for
many applications like two phase flow simulations and phase transitions. In this paper, we develop effective
algorithms so that the solution of the TDGL model can be accurately approximated. Specifically, we
adopt finite element methods for the spatial discretization and study different algorithms for handling
the nonlinear and diffusion terms in the TDGL model. We show that the fully implicit Backward Euler
scheme and the Crank-Nicolson scheme exhibit high accuracy and allow for large time step size. Since
high resolution is needed in the interfacial region when the interfacial thickness is small, we apply newly
developed moving grid deformation techniques [10] to improve both the accuracy and the efficiency. The
advantages of the applied moving grid deformation algorithm over the existing works are highlighted.