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Maximum-norm error analysis of a difference scheme for the space fractional CNLS

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  • Wang, Dongling
  • Xiao, Aiguo
  • Yang, Wei

Abstract

The difference method for the space fractional coupled nonlinear Schrödinger equations (CNLS) is studied. The fractional centered difference is used to approximate the space fractional Laplacian. This scheme conserves the discrete mass and energy. Due to the nonlocal nature of fractional Laplacian, in the classic Sobolev space, it is hard to obtain the error estimation in l∞. To overcome this difficulty, the fractional Sobolev space Hα/2 and a fractional norm equivalence in Hα/2 are introduced. Then the convergence of order O(h2+τ2) in l∞ is proved by fractional Sobolev inequality, where h is the mesh size and τ is the time step. Numerical examples are given to illustrate the theoretical results at last.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:257:y:2015:i:c:p:241-251
    DOI: 10.1016/j.amc.2014.11.026
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    Cited by:

    1. Ding, Hengfei & Tian, Junhong, 2023. "Structure preserving fourth-order difference scheme for the nonlinear spatial fractional Schrödinger equation in two dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 1-18.
    2. Shaojun Chen & Yayun Fu, 2024. "Linearly Implicit Conservative Schemes with a High Order for Solving a Class of Nonlocal Wave Equations," Mathematics, MDPI, vol. 12(15), pages 1-13, August.
    3. 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.
    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. Zhang, Qifeng & Ren, Yunzhu & Lin, Xiaoman & Xu, Yinghong, 2019. "Uniform convergence of compact and BDF methods for the space fractional semilinear delay reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 91-110.
    6. Zhang, Xue & Gu, Xian-Ming & Zhao, Yong-Liang & Li, Hu & Gu, Chuan-Yun, 2024. "Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 462(C).
    7. Fu, Yayun & Hu, Dongdong & Wang, Yushun, 2021. "High-order structure-preserving algorithms for the multi-dimensional fractional nonlinear Schrödinger equation based on the SAV approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 238-255.
    8. Hao, Zhaopeng & Fan, Kai & Cao, Wanrong & Sun, Zhizhong, 2016. "A finite difference scheme for semilinear space-fractional diffusion equations with time delay," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 238-254.
    9. Ran, Yu-Hong & Wang, Jun-Gang & Wang, Dong-Ling, 2015. "On HSS-like iteration method for the space fractional coupled nonlinear Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 482-488.
    10. Wang, Junjie & Xiao, Aiguo, 2019. "Conservative Fourier spectral method and numerical investigation of space fractional Klein–Gordon–Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 348-365.
    11. Wang, Jun-jie & Xiao, Ai-guo, 2018. "An efficient conservative difference scheme for fractional Klein–Gordon–Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 691-709.
    12. 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.

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