A new high-order compact and conservative numerical scheme for the generalized symmetric regularized long wave equations

ABSTRACT

In this paper, a new three-level implicit and fourth-order compact conservative difference scheme is developed for the generalized symmetric regularized long wave equations. The proposed scheme adopts a new time discretization method. The conservation of the new compact scheme is proved, and the discrete conservation laws are obtained. The priori estimation in the maximum norm is discussed by the discrete energy method and mathematical induction. Based on the priori estimation and Brouwer's fixed point theorem, the unique existence of the difference solution is proved. It is proved that the compact difference scheme is unconditionally convergent and stable in the maximum norm, and its convergence order is the fourth order in space and the second order in time. A decoupled and linearized iterative algorithm is proposed to calculate the nonlinear algebraic system generated by the compact scheme, and its convergence is proved. Several numerical experiments are performed to prove the efficiency and reliability of the proposed numerical scheme and to confirm the results obtained from the theoretical analysis.

KEYWORDS:

• GSRLW equations
• compact difference scheme
• discrete conservation laws
• prior estimation
• convergence in $L^{\mathrm{\infty }}$-norm

AMS SUBJECT CLASSIFICATIONS:

• 65N06
• 65M06

Acknowledgments

The authors thank the anonymous reviewers for their valuable suggestions which enhance the quality of this research.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Natural Science Foundation of Liaoning Province of China [grant number 2022-BS-093], the National Natural Science Foundation of China [grant number 42274166] and the Fundamental Research Funds for the Central Universities [grant number 3132022202].