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Maximum-norm error analysis of a conservative scheme for the damped nonlinear fractional Schrödinger equation

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  • Fu, Yayun
  • Song, Yongzhong
  • Wang, Yushun

Abstract

This paper aims to construct a numerical scheme for the damped nonlinear space fractional Schrödinger equation. First, the conservation laws of mass and energy for the continuous equation are derived. Then, based on the fractional centered difference formula, a semi-discrete scheme, which preserves the semi-discrete mass and energy conservation laws is proposed. Further applying the Crank–Nicolson method on the temporal direction gives a fully-discrete conservative scheme. Furthermore, the solvability, boundedness and convergence in the maximum norm of the numerical solutions are given. Some numerical examples are displayed to confirm the theoretical results.

Suggested Citation

  • Fu, Yayun & Song, Yongzhong & Wang, Yushun, 2019. "Maximum-norm error analysis of a conservative scheme for the damped nonlinear fractional Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 206-223.
    • Handle: RePEc:eee:matcom:v:166:y:2019:i:c:p:206-223
    DOI: 10.1016/j.matcom.2019.05.001
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    References listed on IDEAS

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    1. 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.
    2. Manuel Duarte Ortigueira, 2006. "Riesz potential operators and inverses via fractional centred derivatives," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2006, pages 1-12, August.
    3. Liang, Jiarui & Song, Songhe & Zhou, Weien & Fu, Hao, 2018. "Analysis of the damped nonlinear space-fractional Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 495-511.
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