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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)

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.

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*Corresponding author. (Z. Y. Zhang)
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[1] H. Zhang , G.M. Wei and Y. T. Gao , On the general form of the Benjamin-Bona-Mahony equation in fluid mechanics, Czechoslovak J. Phys., 52(3) (2002), pp. 373377.

[3] D. Furihata and T. Matsuo , Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, CRC Press, London, 2010.

[6] S. Koide and D. Furihata , Nonlinear and linear conservative finite difference schemes for regularized long wave equation, Japan J. Indus. Appl. Math., 26(1) (2009), pp. 1540.

[7] T. Matsuo and D. Furihata , Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171(2) (2001), pp. 425447.

[8] T. Yaguchi , T. Matsuo and M. Sugihara , An extension of the discrete variational method to nonuniform grids, J. Comput. Phys., 229(11) (2010), pp. 43824423.

[11] R. E. Bank and D. J. Rose , Some error estimates for the box methods, SIAM J. Numer. Anal., 24(4) (1987), pp. 777787.

[12] W. Hackbusch , On first and second order box schemes, Computing, 41(4) (1989), pp. 277296.

[13] R. H. Li , Z. Y. Chen and W. Wu , Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods, Marcel Dekker Inc., New York, 2000.

[15] Q. X. Wang , Z. Y. Zhang , X. H. Zhang and Q. Y. Zhu , Energy-preserving finite volume element method for the improved Boussinesq equation, J. Comput. Phys., 270 (2014), pp. 5869.

[16] Z. Y. Zhang , Error estimates of finite volume element method for the pollution in groundwater flow, Numer. Methods Partial Differential Equations, 25(2) (2009), pp. 259274.

[17] J. C. Xu and Q. S. Zou , Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numerische Mathematik, 111(3) (2009), pp. 469492.

[18] Z. Y. Zhang and F. Q. Lu , Quadratic finite volume element method for the improved Boussinesq equation, J. Math. Phys., 53(1) (2012), 013505.

[19] D. Dutykh , D. Clamond , P. Milewski and D. Mitsotakis , Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, Euro. J. Appl.Math., 24(5) (2013), pp. 761787.

[20] D. Dutykh , TH. Katsaounis and D. Mitsotakis , Finite volume methods for unidirectional dispersive wave models, Int. J. Numer. Methods Fluids, 71(6) (2013), pp. 717736.

[21] S. Li and L. Vu-Quoc , 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.

[23] A. K. Khalifa , K. R. Raslan and H. M. Alzubaidi , A collocation method with cubic B-splines for solving the MRLW equation, J. Comput. Appl. Math., 212(2) (2008), pp. 406418.

[25] K. R. Raslan , Numerical study of the modified regularized long wave equation, Chaos, Solitons and Fractals, 42(3) (2009), pp. 18451853.

[26] J. X. Cai , A multisymplectic explicit scheme for the modified regularized long-wave equation, J. Comput. Appl. Math., 234(3) (2010), pp. 899905.

[27] M. A. Johnson , On the stability of periodic solutions of the generalized Benjamin-Bona-Mahony equation, Physica D: Nonlinear Phenomena, 239(19) (2010), pp. 18921908.

[28] P. J. Olver , Euler operators and conservation laws of the BBM equation, Mathematical Proceedings of the Cambridge Philosophical Society, 85(1) (1979), pp. 143160.

[29] N. Yi , Y. Huang and H. Liu , 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] M. Dahlby and O. Brynjulf , A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comput., 33(5) (2011), pp. 23182340.

[31] Y. Z. Gong , J. X. Cai and Y. S. Wang , Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys., 279 (2014), pp. 80102.

[32] H. C. Li , J. Q. Sun and M. Z. Qin , Multi-symplectic method for the Zakharov-Kuznetsov equation, Adv. Appl. Math. Mech., 7(1) (2015), pp. 5873.

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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
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