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A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS

Lu Zhang1, Qifeng Zhang2, Hai-wei Sun3   

  1. 1. School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221018, China;
    2. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China;
    3. Department of Mathematics, University of Macau, Macao, China
  • Received:2020-02-05 Revised:2020-05-04 Published:2021-10-15
  • Supported by:
    Q. Zhang was partially supported by Natural Science Foundation of Zhejiang Province (Grant No. LY19A010026), Zhejiang Province “Yucai” Project (2019), Natural Science Foundation of China (Grant No. 11501514) and Fundamental Research Funds of Zhejiang Sci-Tech University (Grant 2019Q072). L. Zhang was partially supported by research from Xuzhou University of Technology (Grant XKY201530) and the “Peiyu” Project from Xuzhou University of Technology (Grant XKY2019104), H. Sun was supported in part by research grants of the Science and Technology Development Fund, Macau SAR (File no. 0118/2018/A3), and MYRG2018-00015-FST from the University of Macau.

Lu Zhang, Qifeng Zhang, Hai-wei Sun. A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS[J]. Journal of Computational Mathematics, 2021, 39(5): 708-732.

This paper focuses on a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations. We firstly establish a three-level finite difference scheme for the time variable followed by the linearized technique of the nonlinear term. Then the fourth-order compact finite difference method is employed to discretize the spatial variables. Hence the accuracy of the discretization is $\mathcal{O}$(τ2 + $h_1^4$ + $h_2^4$) in L2-norm, where τ is the temporal step-size, both h1 and h2 denote spatial mesh sizes in x- and y- directions, respectively. The rigorous theoretical analysis, including the uniqueness, the almost unconditional stability, and the convergence, is studied via the energy argument. Practically, the discretized system holds the block Toeplitz structure. Therefore, the coefficient Toeplitz-like matrix only requires $\mathcal{O}$(M1M2) memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O}$(M1M2(log M1 + log M2)) computational complexity by the fast Fourier transformation, where M1 and M2 denote the numbers of the spatial grids in two different directions. In order to solve the resulting Toeplitz-like system quickly, an efficient preconditioner with the Krylov subspace method is proposed to speed up the iteration rate. Numerical results are given to demonstrate the well performance of the proposed method.

CLC Number: 

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