Conservative Fourier pseudo-spectral schemes for general Klein–Gordon–Schrödinger equations

ABSTRACT

During the past decades, the general Klein–Gordon–Schrödinger systems have been playing more and more important roles in quantum mechanics. In this paper, the conservative Fourier pseudo-spectral schemes are presented for general Klein–Gordon–Schrödinger system. First, we apply the Fourier pseudo-spectral scheme to spatial derivatives, the Crank–Nicolson and leap-frog schemes to Schrödinger and Klein–Gordon equations in time direction, respectively. We find that the scheme can be decoupled and preserve mass and energy conservation laws. Moreover, the stability and convergence of the scheme are discussed, and it is shown that the scheme is of the accuracy $O({\tau }^2+J^{-r})$. However, the scheme is nonlinear. Then, we give linearized scheme of the system. We prove that the scheme can be decoupled, linearized and preserve mass and energy conservation laws. The numerical experiments are given to show the correctness of theoretical results and the efficiency of the schemes.

KEYWORDS:

• Klein–Gordon–Schrödinger equations
• Fourier pseudo-spectral method
• stability
• convergence

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 65M70

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Pu'er University Innovation Team (CXTD003) and Yun Nan Provincial Education Department Foundation Project (2018JS517).