A conservative spectral Galerkin method for the coupled nonlinear space-fractional Schrödinger equations

ABSTRACT

In this paper, a conservative spectral Galerkin method, which is based on the Crank-Nicolson method for the temporal discretization and Legendre spectral Galerkin method for the spatial discretization, is proposed to solve the coupled nonlinear space-fractional Schrödinger equations. We proved that the proposed method satisfies the mass and energy conservation laws in the discrete sense. Moreover, a rigorous analysis of the unique solvability and optimal error estimate in the $L^2$-norm of the Crank-Nicolson spectral Galerkin method are derived. In order to compute the nonlinear system efficiently, we introduce a linear iterative algorithm in implementation. A series of numerical experiments are carried out to illustrate the efficiency of the method.

KEYWORDS:

• Coupled nonlinear space-fractional Schrödinger equations
• Crank-Nicolson method
• Legendre spectral Galerkin method
• mass and energy conservation laws
• spectral accuracy

AMS CLASSIFICATIONS:

• 65M06
• 65M12
• 65M15
• 65M70

Additional information

Funding

This work was supported by National Natural Science Foundation of China (No. 11371289 and No. 11501441).