Numerical analysis of a new conservative scheme for the coupled nonlinear Schrödinger equations

ABSTRACT

We devote the present paper to an efficient conservative scheme for the coupled nonlinear Schrödinger (CNLS) system, based on the Fourier pseudospectral method, the Crank–Nicolson method and leap-frog method. To obtain the present scheme, the key idea consists of two aspects. First, we solve the CNLS system based on its Hamiltonian structure and the resulted scheme can preserve the Hamiltonian nature. Second, we use Fourier pseudospectral method in spatial discretization and Crank–Nicolson/ leap-frog scheme for discretizing linear/ nonlinear terms in time direction, respectively. The proposed scheme is energy-preserving, mass-preserving, uniquely solvable and unconditionally stable, while being decoupled, linearized and suitable for parallel computation in practical computation. Using the energy method and the classical interpolation theory, an error estimate of the proposed scheme is proven strictly without any grid ratio restrictions in the discrete L² norm. Finally, numerical results are reported to verify our theoretical analysis.

KEYWORDS:

• Coupled nonlinear Schrödinger equations
• Fourier pseudospectral method
• Crank–Nicolson
• leap-frog scheme
• error estimate

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M20
• 65M70
• 65T50

Acknowledgements

We are thankful for various comments by the reviewers and editor.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Jialing Wang and Yushun Wang were supported by the National Natural Science Foundation of China [Grant Nos. 11271195,41231173] and the Project of Graduate Education Innovation of Jiangsu Province, China [Grant No. KYLX15-0719].