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An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg–Landau equation

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Abstract

This paper proposes and analyzes a high-order implicit-explicit difference scheme for the nonlinear complex fractional Ginzburg–Landau equation involving the Riesz fractional derivative. For the time discretization, the second-order backward differentiation formula combined with the explicit second-order Gear’s extrapolation is adopted. While for the space discretization, a fourth-order fractional quasi-compact method is used to approximate the Riesz fractional derivative. The scheme is efficient in the sense that, at each time step, only a linear system with a coefficient matrix independent of the time level needs to be solved. Despite of the explicit treatment of the nonlinear term, the scheme is shown to be unconditionally convergent in the \(l^2_h\) norm, semi-\(H^{\alpha /2}_h\) norm and \(l^\infty _h\) norm at the order of \(O(\tau ^2+h^4)\) with \(\tau \) time step and h mesh size. Numerical tests are provided to confirm the accuracy and efficiency of the scheme.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 11771163).

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450000, China

    Pengde Wang

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Chengming Huang

  3. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Chengming Huang

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  1. Pengde Wang
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  2. Chengming Huang
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Correspondence to Chengming Huang.

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Communicated by Jan Hesthaven.

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Wang, P., Huang, C. An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg–Landau equation. Bit Numer Math 58, 783–805 (2018). https://doi.org/10.1007/s10543-018-0698-9

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