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Two New Energy-Preserving Algorithms for Generalized Fifth-Order KdV Equation

  • Qi Hong (a1) (a2), Yushun Wang (a2) and Qikui Du (a2)

In this paper, based on the multi-symplectic formulations of the generalized fifth-order KdV equation and the averaged vector field method, two new energy-preserving methods are proposed, including a new local energy-preserving algorithm which is independent of the boundary conditions and a new global energy-preserving method. We prove that the proposed methods preserve the energy conservation laws exactly. Numerical experiments are carried out, which demonstrate that the numerical methods proposed in the paper preserve energy well.

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*Corresponding author. Email: (Q.Hong), (Y. S.Wang), (Q. K. Du)
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[1] N. Bibi , S. I. A. Tirmizi and S. Haq , Meshless method of lines numerical solution of Kawahara-Type equations, Appl. Math., 2 (2011), pp. 608618.

[2] T. J. Bridges , Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), pp. 147190.

[3] T. J. Bridges and S. Reich , Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), pp. 184193.

[5] J. X. Cai and Y. S. Wang , Local structure-preserving algorithms for the “good” Boussinesq equation, J. Comput. Phys., 239 (2013), pp. 7289.

[6] J. X. Cai , Y. S. Wang and H. Liang , Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schröinger system, J. Comput. Phys., 239 (2013), pp. 3050.

[8] E. Celledoni , V. Grimm , R. I. McLachlan , D. I. McLaren , D. O'Neale , B. Owren and G. R. W. Quispel , Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method, J. Comput. Phys., 231 (2012), pp. 67706789.

[9] E. Celledoni , R. I. McLachlan , D. I. McLaren , B. Owren , G. R. W. Quispel and W. M. Wright , Energy-preserving Runge-Kutta methods, ESAIM: Math. Model. Numer. Anal., 43 (2009), pp. 645649.

[10] Y. Chen , Y. J. Sun and Y. F. Tang , Energy-preserving numerical methods for Landau-Lifshitz equation, J. Phys. A Math. Theor., 44 (2011), 295207.

[11] Y. M. Chen , H. J. Zhu and S. H. Song , Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 181 (2010), pp. 12311241.

[12] Y. F. Cui and D. K. Mao , Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation, J. Comput. Phys., 227 (2007), pp. 376399.

[16] T. Fla , A numerical energy conserving method for the DNLS equation, J. Comput. Phys., 101 (1992), pp. 7179.

[17] D. Furihata , Finite difference schemes for that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), pp. 181205.

[18] 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.

[20] E. Hairer , C. Lubich and G. Wanner , Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Diffenential Equations, Berlin: Springer-Verlag, 2006.

[22] D. Kaya and K. Al-Khaled , A numerical comparision of a Kawahara equation, Phys. Lett. A, 363 (2007), pp. 433439.

[23] J. Marsden , G. Patrick and S. Shkoller , Mulltisymplectic geometry, variational integrators and nonlinear PDEs, Commun. Math. Phys., 199 (1998), pp. 351395.

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

[25] R. I. McLachlan , G. R. W. Quispel and N. Robidoux , Geometric integration using discrete gradients, Philos. Trans. R. Soc. A, 357 (1999), pp. 10211046.

[26] G. R. W. Quispel and D. I. McLaren , A new class of energy-preserving numerical integration methods, J. Phys. A Math. Theor., 41 (2008), 045206.

[27] J. Shen and T. Tang , Spectral and High-order Methods with Applications, Science Press, 2006.

[28] Y. S. Wang , B. Wang and M. Z. Qin , Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A, 51 (2008), pp. 21152136.

[29] J. M. Yuan , J. Shen and J. H. Wu , A Dual-Petrov-Galerkin method for the Kawahara-Type equation, J. Sci. Comput., 34 (2008), pp. 4863.

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