Numerical analysis of a conservative linear compact difference scheme for the coupled Schrödinger–Boussinesq equations

ABSTRACT

In this article, a decoupled and linearized compact difference scheme is investigated to solve the coupled Schrödinger–Boussinesq equations numerically. We establish the convergence rates for the error at the order of $O({\tau }^2+h^4)$ in the $l^2$-norm with the time step τ and mesh size h. The linear scheme is proved to conserve the total energy which is defined as a recursion relationship. Due to the difficulty in obtaining the priori estimate from the discrete energy, we utilize cut-off function technique to prove the convergence. The numerical results are reported to verify the theoretical analysis, and the numerical comparison between our scheme with previous methods are conducted to show the efficiency of our scheme.

KEYWORDS:

• Schrödinger–Boussinesq equations
• compact difference scheme
• cut-off function technique
• conservative law
• convergence

AMS Classifications:

• 65M06
• 35Q55
• 35Q51

Acknowledgments

The authors are very grateful to both reviewers for carefully reading the paper and for their comments and suggestions which have led to improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by Jiangsu Innovation Program for Graduate Education (2016) under [grant number KYZZ16_0161] and the National Science Foundation of China under [grant number 11571181].