Volume 39, Issue 5
A Fast Compact Difference Method for Two-Dimensional Nonlinear Space-Fractional Complex Ginzburg-Landau Equations

J. Comp. Math., 39 (2021), pp. 708-732.

Published online: 2021-08

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• Abstract

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 $\mathcal{O}(\tau^2+h_1^4+h_2^4)$ in $L_2$-norm, where $\tau$ is the temporal step-size, both $h_1$ and $h_2$ 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 $\mathcal{O} \big( M_{1}M_{2} \big)$ memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O} \big( M_{1}M_{2} (\log M_{1}+\log M_{2})\big)$ computational complexity by the fast Fourier transformation, where $M_1$ and $M_2$ 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.

• Keywords

Space-fractional Ginzburg-Landau equation, Compact scheme, Boundedness, Convergence, Preconditioner, FFT.

26A33, 35R11, 65M06, 65M1

yulu7517@126.com (Lu Zhang)

zhangqifeng0504@163.com (Qifeng Zhang)

hsun@um.edu.mo (Hai-Wei Sun)

• BibTex
• RIS
• TXT
@Article{JCM-39-708, author = {Zhang , Lu and Zhang , Qifeng and Sun , Hai-Wei}, title = {A Fast Compact Difference Method for Two-Dimensional Nonlinear Space-Fractional Complex Ginzburg-Landau Equations}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {5}, pages = {708--732}, abstract = {

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 $\mathcal{O}(\tau^2+h_1^4+h_2^4)$ in $L_2$-norm, where $\tau$ is the temporal step-size, both $h_1$ and $h_2$ 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 $\mathcal{O} \big( M_{1}M_{2} \big)$ memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O} \big( M_{1}M_{2} (\log M_{1}+\log M_{2})\big)$ computational complexity by the fast Fourier transformation, where $M_1$ and $M_2$ 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2005-m2020-0029}, url = {http://global-sci.org/intro/article_detail/jcm/19378.html} }
TY - JOUR T1 - A Fast Compact Difference Method for Two-Dimensional Nonlinear Space-Fractional Complex Ginzburg-Landau Equations AU - Zhang , Lu AU - Zhang , Qifeng AU - Sun , Hai-Wei JO - Journal of Computational Mathematics VL - 5 SP - 708 EP - 732 PY - 2021 DA - 2021/08 SN - 39 DO - http://doi.org/10.4208/jcm.2005-m2020-0029 UR - https://global-sci.org/intro/article_detail/jcm/19378.html KW - Space-fractional Ginzburg-Landau equation, Compact scheme, Boundedness, Convergence, Preconditioner, FFT. AB -

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 $\mathcal{O}(\tau^2+h_1^4+h_2^4)$ in $L_2$-norm, where $\tau$ is the temporal step-size, both $h_1$ and $h_2$ 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 $\mathcal{O} \big( M_{1}M_{2} \big)$ memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O} \big( M_{1}M_{2} (\log M_{1}+\log M_{2})\big)$ computational complexity by the fast Fourier transformation, where $M_1$ and $M_2$ 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.

Lu Zhang, Qifeng Zhang & Hai-Wei Sun. (2021). A Fast Compact Difference Method for Two-Dimensional Nonlinear Space-Fractional Complex Ginzburg-Landau Equations. Journal of Computational Mathematics. 39 (5). 708-732. doi:10.4208/jcm.2005-m2020-0029
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