A linearized Crank-Nicolson-Galerkin method for the Ginzburg-Landau model

The time-dependent Ginzburg-Landau model is used extensively in studying the nonequilibrium state of superconductivity. The computer simulation of this model requires highperformance computing power and reliable and efficient numerical methods to solve the Ginzburg-Landau equations. In this paper, a linearized Crank-Nicolson-Galerkin method is proposed for solving these nonlinear and coupled partial differential equations. The method uses the Galerkin finite element approximation in spatial discretization and the Crank-Nicolson implicit scheme in time discretization, together with certain techniques which linearize and decouple the Ginzburg-Landau equations. While retaining the stability and accuracy of the Crank-Nicolson scheme, the proposed approach results in symmetric and positive definite matrix problems, thus substantially improving the computational efficiency. Furthermore, and even more important, the proposed approach is suitable for large-scale parallel computation. Numerical results from simulating the vortex dynamics of superconductivity by using the linearized Crank-Nicolson-Galerkin method are presented.