Skip to main content
×
×
Home

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

  • Jinliang Yan (a1) (a2), Ming-Chih Lai (a3), Zhilin Li (a4) and Zhiyue Zhang (a1)
Abstract
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.

Copyright
Corresponding author
*Corresponding author. Email: zhangzhiyue@njnu.edu.cn (Z. Y. Zhang)
References
Hide All
[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. 373377.
[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. 231244.
[3] Furihata D. and Matsuo T., Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, CRC Press, London, 2010.
[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. 405406.
[5] Durán A. and López-Marcos M. A., Conservative numerical methods for solitary wave interactions, J. Phys. A. Math. Theor., 36(28) (2003), pp. 77617770.
[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. 1540.
[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. 425447.
[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. 43824423.
[9] Matsuo T. and Kuramae H., An alternating discrete variational derivative method, AIP Conference Proceedings, 1479(1) (2012), pp. 12601263.
[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. 2932.
[11] Bank R. E. and Rose D. J., Some error estimates for the box methods, SIAM J. Numer. Anal., 24(4) (1987), pp. 777787.
[12] Hackbusch W., On first and second order box schemes, Computing, 41(4) (1989), pp. 277296.
[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.
[14] Li Y. and Li R. H., Generalized difference methods on arbitrary quadrilateral networks, J. Comput. Math., 17(6) (1999), pp. 653672.
[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. 5869.
[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. 259274.
[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. 469492.
[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.
[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. 761787.
[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. 717736.
[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. 18391875.
[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. 183193.
[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. 406418.
[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. 346354.
[25] Raslan K. R., Numerical study of the modified regularized long wave equation, Chaos, Solitons and Fractals, 42(3) (2009), pp. 18451853.
[26] Cai J. X., A multisymplectic explicit scheme for the modified regularized long-wave equation, J. Comput. Appl. Math., 234(3) (2010), pp. 899905.
[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. 18921908.
[28] Olver P. J., Euler operators and conservation laws of the BBM equation, Mathematical Proceedings of the Cambridge Philosophical Society, 85(1) (1979), pp. 143160.
[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. 351366.
[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. 23182340.
[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. 80102.
[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. 5873.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 32 *
Loading metrics...

Abstract views

Total abstract views: 243 *
Loading metrics...

* Views captured on Cambridge Core between 9th January 2017 - 18th January 2018. This data will be updated every 24 hours.