A three-level finite difference method with preconditioning technique for two-dimensional nonlinear fractional complex Ginzburg–Landau equations

Abstract

In the paper, we study two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equations. A centered finite difference method is exploited to discretize the spatial variables, while a three-level finite difference scheme is applied for the time integration. Theoretically, we prove the proposed method is uniquely solvable and unconditionally stable, with second order accuracy on both time and space, respectively. As the resulting discretized systems possess the block-Toeplitz structure, we proposed the preconditioned GMRES method with a block circulant preconditioner to speed up the convergence rate of the iteration. Meanwhile, fast Fourier transformation is utilized to reduce the complexity for calculating the discretized systems. Numerical experiments are carried out to verify the theoretical results and demonstrate that the proposed method enjoys the excellent computational advantage.