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A radial basis function-Hermite finite difference (RBF-HFD) method for the cubic-quintic complex Ginzburg–Landau equation

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Abstract

In this paper, the cubic–quintic complex Ginzburg–Landau (CQCGL) equation is numerically studied in 1D, 2D and 3D spaces. First, by the Strang splitting technique, the CQCGL equation is decomposed into three subproblems. After spatial discretization, the first and third problems lead to nonlinear ODEs that are solved by the Runge–Kutta technique. For the second problem, involving the spatial derivative, a fourth-order RBF—generated Hermite finite difference (RBF–HFD) method is used. The RBF–HFD scheme improves the accuracy in space direction compared with RBF-FD method, but temporal convergence order remains two due to the use of Strang splitting and Crank–Nicolson schemes in time direction. To improve the order of convergence in time direction, a temporal Richardson extrapolation technique is applied. Numerical results show that the order of convergence is improved from \(O({\tau ^2} + {h^4})\) to \(O({\tau ^4} + {h^4})\). Numerical tests are provided to confirm the accuracy and efficiency of the proposed method.

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Acknowledgements

The authors thank the reviewers for their useful comments and suggestions that improved the paper.

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Authors and Affiliations

  1. School of Mathematics, Shahid Rajaee Teacher Training University, Lavizan, Tehran, 16788, Iran

    Majid Haghi

  2. Department of Applied Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran

    Mohammad Ilati

  3. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), No. 424 Hafez Ave., Tehran, Iran

    Mehdi Dehghan

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  1. Majid Haghi
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  2. Mohammad Ilati
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Correspondence to Mohammad Ilati.

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Haghi, M., Ilati, M. & Dehghan, M. A radial basis function-Hermite finite difference (RBF-HFD) method for the cubic-quintic complex Ginzburg–Landau equation. Comp. Appl. Math. 42, 115 (2023). https://doi.org/10.1007/s40314-023-02256-3

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  • DOI: https://doi.org/10.1007/s40314-023-02256-3

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