Uniform error estimates of fourth-order conservative linearized difference scheme for a mathematical model for long wave

Abstract

This article is devoted to the study of high-order conservative linearized finite difference scheme for a model of nonlinear dispersive equation: regularized long-wave-Korteweg–de Vries equation. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergence of the difference scheme is proved by using the energy method to be of fourth-order in space and second-order in time in the discrete $L^{\mathrm{\infty }}$-norm. An application on the regularized long-wave equation as well for the modified regularized long-wave equation are discussed numerically in detail. Furthermore, interaction of solitary waves with different amplitudes is shown. The three invariants of the motion are evaluated to determine the conservation properties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied for both equations. Some numerical results and comparisons with other existing methods in the literature are presented. The numerical results show that the linearized difference scheme of this article improves the accuracy of the space and time direction and shortens computation time largely.

Keywords:

• Regularized long-wave-Korteweg–de Vries equation
• conservation
• solvability
• high-order difference scheme
• stability
• convergence
• solitary waves

2000 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M15

Acknowledgments

We would like to thank the reviewers that their comments and suggestions have really improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.