Conservative compact difference scheme for the generalized Korteweg-de Vries equation

Abstract

In this paper, a linear mass and energy conservative finite difference scheme for the generalized Korteweg-de Vries (GKdV) equation is proposed and analyzed. The scheme is three-level and linear implicit and gives second- and fourth-order accuracy in time and space, respectively. It is rigorously proved by using the discrete energy method that the proposed difference scheme is uniquely solvable, stable and convergent. Numerical examples are given to confirm the stability and convergence of the numerical solution with fourth-order accuracy and the effectiveness of the present scheme for handling the single and multi-solitary waves for a long time.

KEYWORDS:

• GKdV equation
• multi-solitary waves
• stability
• finite difference scheme
• conservation

2010 AMS subject classifications:

• 35Q53
• 65M06

Acknowledgments

The authors also thank the reviewers and editors for their constructive comments and suggestions which significantly improved the quality of this paper.

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 Fujian Province, China (No: 2020J01796).