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New Conservative Finite Volume Element Schemes for the Modified Regularized Long Wave Equation

Part of: Hamiltonian and Lagrangian mechanics Waves Numerical methods Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  09 January 2017

Jinliang Yan ,
Ming-Chih Lai ,
Zhilin Li  and
Zhiyue Zhang
Jinliang Yan
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China Wuyi University, Department of Mathematics and Computer, Wuyishan, Fujian 354300, China
Ming-Chih Lai
Affiliation:
Department of Applied mathematics, National Chiao Tung University, Taiwan
Zhilin Li
Affiliation:
Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205, USA
Zhiyue Zhang*
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China
*
*Corresponding author. Email:zhangzhiyue@njnu.edu.cn (Z. Y. Zhang)
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Abstract

In this paper, we propose a new energy-preserving scheme and a new momentum-preserving scheme for the modified regularized long wave equation. The proposed schemes are designed by using the discrete variational derivative method and the finite volume element method. For comparison, we also propose a finite volume element scheme. The conservation properties of the proposed schemes are analyzed and we find that the energy-preserving scheme can precisely conserve the discrete total mass and total energy, the momentum-preserving scheme can precisely conserve the discrete total mass and total momentum, while the finite volume element scheme merely conserve the discrete total mass. We also analyze their linear stability property using the Von Neumann theory and find that the proposed schemes are unconditionally linear stable. Finally, we present some numerical examples to illustrate the effectiveness of the proposed schemes.

Keywords

Energymomentummassfinite volume element methodMRLW equation

MSC classification

Secondary: 65M99: None of the above, but in this section 70H06: Completely integrable systems and methods of integration 74J35: Solitary waves 74S99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 2 , April 2017 , pp. 250 - 271
Copyright
Copyright © Global-Science Press 2017 

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