An averaged vector field Legendre spectral element method for the nonlinear Schrödinger equation

ABSTRACT

In this paper, we propose an averaged vector field Legendre spectral element (AVFLSE) method for the nonlinear Schrödinger (NLS) equation. The key idea is to rewrite the NLS equation as an infinite-dimensional Hamiltonian PDE and discrete the Hamiltonian PDE by using the Legendre spectral element (LSE) method in space and an averaged vector field (AVF) method in time. We show that applying a Galerkin method to Hamiltonian PDEs in space can lead to a semi-discrete system which can be cast into Hamiltonian ODEs. For the NLS equation, the concrete canonical form of the resulting Hamiltonian ODEs is presented. This methodology can ensure that the structure matrix of the resulting Hamiltonian ODEs is sparse and the full discrete scheme for the Hamiltonian PDEs is an energy-preserving, unconditionally linearly stable and symmetric method. With the aid of cut-off technique, we also derive the error estimate and the new method turns out to be convergent with the convergence order of $\mathcal{O}(\mathrm{\Delta }{t}^2+\mathrm{\Delta }{x}^{min(N,r)}{N}^{-r})$ in the discrete $L^2$-norm with the exact solution $u\in H^r$, where N is the order of the Legendre cardinal basis functions. Numerical experiments are provided to demonstrate the energy-preserving property and the convergence behaviour of the method.

KEYWORDS:

• Hamiltonian system
• energy-preserving
• nonlinear Schrödinger equation
• AVFSE method
• Legendre spectral element method
• averaged vector field method
• error estimate

2010 AMS CLASSIFICATIONS:

• 37K05
• 65M12
• 65M15
• 65M20
• 65M70

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Jiangsu Collaborative Innovation Center for Climate Change, the National Natural Science Foundation of China (Grant Nos. 11271195, 41231173) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.