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Exponential Runge–Kutta Method for Two-Dimensional Nonlinear Fractional Complex Ginzburg–Landau Equations

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Abstract

In this work, we study numerically two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equations. A centered finite difference method is exploited to discretize the spatial variables and leads to a system of the ordinary differential equation, in which the resulting coefficient matrix is complex symmetric and possesses the block Toeplitz structure. An exponential Runge–Kutta method is employed to solve such a system of the ordinary differential equation. Theoretically, the proposed method is second-order accuracy in space and fourth-order accuracy in time, respectively. In the practical implementation, the product of a block Toeplitz matrix exponential and a vector is calculated by the shift-invert Lanczos method. Meanwhile, the sectorial operator (the coefficient matrix) guarantees the fast approximation by the shift-invert Lanczos method. Numerical experiments are carried out to testify the theoretical results and demonstrate that the proposed method enjoys the excellent computational advantage.

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Acknowledgements

The authors are very grateful to the anonymous referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou, 221018, Jiangsu, China

    Lu Zhang

  2. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Qifeng Zhang

  3. Department of Mathematics, University of Macau, Macao, China

    Hai-Wei Sun

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  1. Lu Zhang
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Correspondence to Hai-Wei Sun.

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The work was supported in part by Natural Science Foundation of China (Grant Number 11501514), Natural Sciences Foundation of Zhejiang Province (Grant Number LY19A010026), the Zhejiang Province “Yucai” Project, the Fundamental Research Funds of Zhejiang Sci-Tech University (Grant Number 2019Q072), the research from Xuzhou University of Technology (Grant XKY2015302), the “Peiyu” Project from Xuzhou University of Technology (Grant Number XKY2019104), the Science and Technology Development Fund, Macau SAR (File Number 0118/2018/A3), and MYRG2018-00015-FST from University of Macau.

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Zhang, L., Zhang, Q. & Sun, HW. Exponential Runge–Kutta Method for Two-Dimensional Nonlinear Fractional Complex Ginzburg–Landau Equations. J Sci Comput 83, 59 (2020). https://doi.org/10.1007/s10915-020-01240-x

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