Preconditioned fourth-order exponential integrator for two-dimensional nonlinear fractional Ginzburg-Landau equation

In this work, we present a high-order numerical method for the two-dimensional nonlinear space-fractional complex Ginzburg-Landau equation (FCGLE). Firstly, a fourth-order approximation is adopted to discretize the spatial Riesz fractional derivatives that leads to a semi-linear system of ordinary differential equations (ODEs), whose coefficient matrix has the complex block Toeplitz structure. Then a fourth-order exponential integrator method is used to solve the corresponding semi-linear ODEs system. In light of the results in theory, the proposed algorithm is fourth-order accuracy in both time and space. In the specific implementation of the proposed algorithm, due to the special structure of the coefficient matrix, the products of some phi-functions of matrices (related to the matrix exponential) and vectors are computed by the shift-invert Lanczos technique in the exponential integrator. In order to calculate the linear system of equations arising from the shift-invert Lanczos procedure, two classes of efficient preconditioners including Strang's circulant preconditioner/tau matrix preconditioner are constructed and implemented by fast Fourier transform and fast sine transform, respectively. Numerical examples with and without exact solutions are implemented to confirm the effectiveness of the current algorithm.