New conservative finite volume element schemes for the modified Korteweg–de Vries equation

Abstract

In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg–de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd.

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

“…However, there was no analysis of convergence in this research [2]. Yan proposed three conservative finite volume element schemes based on the discrete variational derivative method [26], but there was no analysis of convergence. Winebery developed an implicit-stepping scheme for KdV equation in temporal direction and spectral methods in space [25].…”

A new conservative finite difference scheme for the generalized Rosenau–KdV–RLW equation

“…Nonlinear wave phenomena play an important role in engineering and sciences. In the past, many scientists have studied about different mathematical models to explain the wave behavior, such as the KdV equation (Korteweg and Vries 1895;Ozer and Kutluay 2005;Skogestad and Kalisch 2009;Kim et al 2012;Yan et al 2016), the Rosenau equation (Rosenau 1986(Rosenau , 1988Park 1992), the Rosenau-KdV equation (Zuo 2009;Esfahani 2011;Triki and Biswas 2013;Zheng and Zhou 2014), the Rosenau-RLW equation (Pan and Zhang 2012;Wongsaijai et al , 2019, and many others (Lu and Chen 2015;Coclite and Ruvob 2017;Mohanty and Kaur 2019;Kaur and Mohanty 2019).…”