Fast iterative solvers for the two-dimensional spatial fractional Ginzburg-Landau equations

Abstract In this work, we propose an alternating direction implicit (ADI) scheme to discrete the two-dimensional spatial fractional Ginzburg-Landau equations. Meanwhile, a matrix splitting iteration method that preserves the Toeplitz structure is proposed for solving the resulting complex linear system, in which the circulant preconditioning technique and fast Fourier transform(FFT) can be adopted to improve computing efficiency. Convergence properties of the corresponding method are derived under some conditions. Numerical experiments demonstrate that our method outperforms some existing iteration methods in iterative steps and computing time.