Analysis of a Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger equation

ABSTRACT

In this paper, we focus on constructing and analysing a new Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger (KGS) equation. After rewriting the KGS equation as an infinite-dimensional Hamiltonian system, we use a Fourier pseudo-spectral method to discrete the system in space to obtain a semi-discrete system, which can be cast into a canonical finite-dimensional Hamiltonian form. Then, an energy-preserving and charge-preserving scheme is constructed by using the symmetric discrete gradient method. Based on the discrete conservation laws and the equivalence of the semi-norm between the Fourier pseudo-spectral method and the finite difference method, the pseudo-spectral solution of the proposed scheme is proved to be bounded in the discrete $L_{\mathrm{\infty }}$ norm. The proposed scheme is shown to be convergent with the convergence order of $\mathcal{O}(J^{-r}+{\tau }^2)$ in the discrete $L_2$ norm afterwards, where J is the number of nodes and τ is the time step size. Numerical experiments are conducted to verify the theoretical analysis.

Keywords:

• Fourier pseudo-spectral method
• energy-preserving scheme
• canonical finite-dimensional Hamiltonian form
• Klein–Gordon–Schrödinger equation
• discrete gradient method

2010 AMS SUBJECT CLASSIFICATIONS:

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

Acknowledgements

J. L. Wang would also thank the Department of Mathematics $\mathrm{\&}$ Statistics at York University for hosting her visit.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

J. L. Wang and Y. S. Wang were supported by the National Natural Science Foundation of China, Jiangsu Collaborative Innovation Center for Climate Change, China [under Grant Nos. 41231173 and 11271195], the Priority Academic Program Development of Jiangsu Higher Education Institutions, China and the Project of Graduate Education Innovation of Jiangsu Province, China [Grant No. KYLX15-0719]. D. Liang was supported by Natural Sciences and Engineering Research Council of Canada.