Abstract
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.
Highlights
The space fractional Schrödinger equation (FSE) is a natural extension of the classic Schrödinger equation, and it has been successfully used to describe the fractional quantum phenomena
For the well-posedness, global attractor, soliton dynamics and ground states related to the FSE, we refer to Refs. [5,6,7] and the references therein
The main objective of this paper is to develop a linearized Galerkin–Legendre spectral scheme for solving the strongly coupled fractional Schrödinger equations (SCFSEs)
Summary
The space fractional Schrödinger equation (FSE) is a natural extension of the classic Schrödinger equation, and it has been successfully used to describe the fractional quantum phenomena. Li et al [21,22,23] investigated a series of Galerkin finite element methods for the FSE, and they discussed the conservation, well-posedness and convergence properties of the discrete systems. We intend to consider the unconditionally convergent spectral method, which takes advantage of spectral accuracy in space Based on these considerations, the main objective of this paper is to develop a linearized Galerkin–Legendre spectral scheme for solving the SCFSEs. The derived scheme can preserve both the mass- and the energy-conservation laws in the discrete sense. The discrete scheme is proved to be unconditionally convergent with second-order accuracy in time and spectral accuracy in space by the energy method. For convenience of theoretical analysis, one can define the following semi-norm and norm:.
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