A novel energy-preserving scheme for the coupled nonlinear Schrödinger equations

ABSTRACT

This paper explores an efficient energy-preserving scheme for the coupled nonlinear Schrödinger system with emphasis on preserving the Hamiltonian structure based on the weak formulation of the system. First, the Schrödinger system is discretized in spatial direction by the Galerkin spectral element method, and the resulting semi-discrete system is rewritten as a finite-dimensional canonical Hamiltonian system. Second, we apply the energy method to discretize the resulting ordinary differential equations in time direction and obtain an efficient conservative scheme. Using the energy method, an error estimate without any restriction on the grid ratio is derived, and the new scheme is convergent with the order $\mathcal{O}({\tau }^2+h^2)$ in the discrete $L^2$-norm, where τ and h denote the time and the space steps, respectively. Finally, comprehensive numerical results including comparisons with the semi-explicit multi-symplectic Euler-box and the Crank–Nicolson finite difference method are given to illustrate the superiority of the proposed energy-preserving scheme.

KEYWORDS:

• Coupled nonlinear Schrödinger equations
• energy-preserving scheme
• Galerkin spectral element method
• energy method
• error estimate

2010 MSC SUBJECT CLASSIFICATION:

• 65M06
• 65M20
• 65M70

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This study is supported by the National Natural Science Foundation of China [Grant Nos. 41231173 and 11771213], and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.