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An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator

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  • Almushaira, Mustafa

Abstract

In this study, we investigate a high-order accurate conservative finite difference scheme by utilizing a fourth-order fractional central finite difference method for the two-dimensional Riesz space-fractional nonlinear Schrödinger wave equation. The conservation laws of the discrete difference scheme are shown. Meanwhile, the exactness, uniqueness, and prior estimate of the numerical solution are rigorously established. Then, it is proved that the proposed scheme is unconditionally convergent in the discrete L2 and Hγ/2 norm, where γ is a fractional order. Furthermore, we demonstrate that when the fractional order γ and the spatial grid number J increase, the block-Toeplitz coefficient matrix generated by the spatial discretization becomes ill-conditioned. As a result, we adopt an effective linearized iteration method for the nonlinear system, allowing it to be solved efficiently by the Krylov subspace solver with an appropriate circulant preconditioner, in which the fast Fourier transform is applied to speed up the computational cost at each iterative step. Finally, numerical experiments are presented to validate the theoretical findings and the efficiency of the fast algorithm.

Suggested Citation

  • Almushaira, Mustafa, 2023. "An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 424-447.
    • Handle: RePEc:eee:matcom:v:210:y:2023:i:c:p:424-447
    DOI: 10.1016/j.matcom.2023.03.019
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    References listed on IDEAS

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    1. Xing, Zhiyong & Wen, Liping, 2019. "Numerical analysis and fast implementation of a fourth-order difference scheme for two-dimensional space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 155-166.
    2. Wang, Dongling & Xiao, Aiguo & Yang, Wei, 2015. "Maximum-norm error analysis of a difference scheme for the space fractional CNLS," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 241-251.
    3. Chen, S. & Liu, F. & Jiang, X. & Turner, I. & Anh, V., 2015. "A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 591-601.
    4. Li, Meng & Zhao, Yong-Liang, 2018. "A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 758-773.
    5. Mustafa Almushaira & Fei Liu, 2020. "Fourth-order time-stepping compact finite difference method for multi-dimensional space-fractional coupled nonlinear Schrödinger equations," Partial Differential Equations and Applications, Springer, vol. 1(6), pages 1-29, December.
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