A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

Advances in Difference Equations

article

Fei, Mingfa¹  Zhang, Guoyu⁴  Wang, Nan⁵  Huang, Chengming⁶ 

[1] School of Computer Engineering and Applied Mathematics, Changsha University;College of Arts and Sciences, National University of Defense Technology;Hunan Province Key Laboratory of Industrial Internet Technology and Security, Changsha University;School of Mathematical Sciences, Inner Mongolia University;School of Mathematics and Statistics, Zhengzhou University;School of Mathematics and Statistics, Huazhong University of Science and Technology;Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology

关键词: Fractional Schrödinger equation;    Legendre spectral method;    Conservation law;    Unconditional convergence;    Spectral accuracy;   

DOI  :  10.1186/s13662-020-03017-w

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

In this paper, based on Galerkin–Legendre spectral method for space discretization and a linearized Crank–Nicolson difference scheme in time, a fully discrete spectral scheme is developed for solving the strongly coupled nonlinear fractional Schrödinger equations. We first prove that the proposed scheme satisfies the conservation laws of mass and energy in the discrete sense. Then a prior bound of the numerical solutions in$L^{\infty }$ -norm is obtained, and the spectral scheme is shown to be unconditionally convergent in$L^{2}$ -norm, with second-order accuracy in time and spectral accuracy in space. Finally, some numerical results are provided to validate our theoretical analysis.

【 授权许可】

CC BY