An efficient Chebyshev-Tau spectral method for Ginzburg-Landau-Schrödinger equations

We propose an efficient time-splitting Chebyshev-Tau spectral method for the Ginzburg-Landau-Schrödinger equation with zero/nonzero far-field boundary conditions. The key technique that we apply is splitting the Ginzburg-Landau-Schrödinger equation in time into two parts, a nonlinear equation and a linear equation. The nonlinear equation is solved exactly; while the linear equation in one dimension is solved with Chebyshev-Tau spectral discretization in space and Crank-Nicolson method in time. The associated discretized system can be solved very efficiently since they can be decoupled into two systems, one for the odd coefficients, the other for the even coefficients. The associated matrices have a quasi-tridiagonal structure which allows a direction solution to be obtained. The computation cost of the method in one dimension is O(Nlog(N)) compared with that of the non-optimized one, which is O(N²). By applying the alternating direction implicit (ADI) technique, we extend this efficient method to solve the Ginzburg-Landau-Schrödinger equation both in two dimensions and in three dimensions, respectively. Numerical accuracy tests of the method in one dimension, two dimensions and three dimensions are presented. Application of the method to study the semi-classical limits of Ginzburg-Landau-Schrödinger equation in one dimension and the two-dimensional quantized vortex dynamics in the Ginzburg-Landau-Schrödinger equation are also presented.