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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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References

[1] Zhang, H., Wei, G.M. and Gao, Y. T., On the general form of the Benjamin-Bona-Mahony equation in fluid mechanics, Czechoslovak J. Phys., 52(3) (2002), pp. 373–377.CrossRefGoogle Scholar
[2] Karakoc, S. B. G., Geyikli, T. and Bashan, A., A numerical solution of the modified regularized long wave (MRLW) equation using quartic B-splines, TWMS J. Eng. Math., 3(2) (2013), pp. 231–244.Google Scholar
[3] Furihata, D. and Matsuo, T., Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, CRC Press, London, 2010.Google Scholar
[4] Furihata, D. and Mori, M., A stable finite difference scheme for the Cahn-Hilliard equation based on a Lyapunov functional, J. Appl. Math. Mech., 76(1) (1996), pp. 405–406.Google Scholar
[5] Durán, A. and López-Marcos, M. A., Conservative numerical methods for solitary wave interactions, J. Phys. A. Math. Theor., 36(28) (2003), pp. 7761–7770.Google Scholar
[6] Koide, S. and Furihata, D., Nonlinear and linear conservative finite difference schemes for regularized long wave equation, Japan J. Indus. Appl. Math., 26(1) (2009), pp. 15–40.Google Scholar
[7] Matsuo, T. and Furihata, D., Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171(2) (2001), pp. 425–447.Google Scholar
[8] Yaguchi, T., Matsuo, T. and Sugihara, M., An extension of the discrete variational method to nonuniform grids, J. Comput. Phys., 229(11) (2010), pp. 4382–4423.CrossRefGoogle Scholar
[9] Matsuo, T. and Kuramae, H., An alternating discrete variational derivative method, AIP Conference Proceedings, 1479(1) (2012), pp. 1260–1263.CrossRefGoogle Scholar
[10] Kuramae, H. and Matsuo, T., An alternating discrete variational derivative method for coupled partial differential equations, Japan Soc. Indus. Appl. Math. Lett., 4 (2012), pp. 29–32.Google Scholar
[11] Bank, R. E. and Rose, D. J., Some error estimates for the box methods, SIAM J. Numer. Anal., 24(4) (1987), pp. 777–787.CrossRefGoogle Scholar
[12] Hackbusch, W., On first and second order box schemes, Computing, 41(4) (1989), pp. 277–296.Google Scholar
[13] Li, R. H., Chen, Z. Y. and Wu, W., Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods, Marcel Dekker Inc., New York, 2000.CrossRefGoogle Scholar
[14] Li, Y. and Li, R. H., Generalized difference methods on arbitrary quadrilateral networks, J. Comput. Math., 17(6) (1999), pp. 653–672.Google Scholar
[15] Wang, Q. X., Zhang, Z. Y., Zhang, X. H. and Zhu, Q. Y., Energy-preserving finite volume element method for the improved Boussinesq equation, J. Comput. Phys., 270 (2014), pp. 58–69.Google Scholar
[16] Zhang, Z. Y., Error estimates of finite volume element method for the pollution in groundwater flow, Numer. Methods Partial Differential Equations, 25(2) (2009), pp. 259–274.Google Scholar
[17] Xu, J. C. and Zou, Q. S., Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numerische Mathematik, 111(3) (2009), pp. 469–492.Google Scholar
[18] Zhang, Z. Y. and Lu, F. Q., Quadratic finite volume element method for the improved Boussinesq equation, J. Math. Phys., 53(1) (2012), 013505.Google Scholar
[19] Dutykh, D., Clamond, D., Milewski, P. and Mitsotakis, D., Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, Euro. J. Appl.Math., 24(5) (2013), pp. 761–787.Google Scholar
[20] Dutykh, D., Katsaounis, TH. and Mitsotakis, D., Finite volume methods for unidirectional dispersive wave models, Int. J. Numer. Methods Fluids, 71(6) (2013), pp. 717–736.Google Scholar
[21] Li, S. and Vu-Quoc, L., Finite difference calculas invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 32(6) (1995), pp. 1839–1875.CrossRefGoogle Scholar
[22] Gardner, L. R. T., Gardner, G. A., Ayoub, F. A. and Amein, N. K., Approximations of solitary waves of the MRLW equation by B-spline finite element, Arabian J. Sci. Eng. A Sci., 22(2) (1997), pp. 183–193.Google Scholar
[23] Khalifa, A. K., Raslan, K. R. and Alzubaidi, H. M., A collocation method with cubic B-splines for solving the MRLW equation, J. Comput. Appl. Math., 212(2) (2008), pp. 406–418.Google Scholar
[24] Khalifa, A. K., Raslan, K. R. and Alzubaidi, H. M., A finite difference scheme for the MRLW and solitary wave interactions, Appl. Math. Comput., 189(1) (2007), pp. 346–354.Google Scholar
[25] Raslan, K. R., Numerical study of the modified regularized long wave equation, Chaos, Solitons and Fractals, 42(3) (2009), pp. 1845–1853.Google Scholar
[26] Cai, J. X., A multisymplectic explicit scheme for the modified regularized long-wave equation, J. Comput. Appl. Math., 234(3) (2010), pp. 899–905.Google Scholar
[27] Johnson, M. A., On the stability of periodic solutions of the generalized Benjamin-Bona-Mahony equation, Physica D: Nonlinear Phenomena, 239(19) (2010), pp. 1892–1908.Google Scholar
[28] Olver, P. J., Euler operators and conservation laws of the BBM equation, Mathematical Proceedings of the Cambridge Philosophical Society, 85(1) (1979), pp. 143–160.Google Scholar
[29] Yi, N., Huang, Y. and Liu, H., A direct discontinuous Galerkin method for the generallized Korteweg-de Vries equation: energy conservation and boundary effect, J. Comput. Phys., 242 (2013), pp. 351–366.Google Scholar
[30] Dahlby, M. and Brynjulf, O., A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comput., 33(5) (2011), pp. 2318–2340.CrossRefGoogle Scholar
[31] Gong, Y. Z., Cai, J. X. and Wang, Y. S., Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys., 279 (2014), pp. 80–102.Google Scholar
[32] Li, H. C., Sun, J. Q. and Qin, M. Z., Multi-symplectic method for the Zakharov-Kuznetsov equation, Adv. Appl. Math. Mech., 7(1) (2015), pp. 58–73.Google Scholar