Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T18:41:50.772Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  11 July 2017

Qi Hong ,
Yushun Wang  and
Qikui Du
Qi Hong*
Affiliation:
Graduate School of China Academy of Engineering Physics, Beijing 100088, China Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Jiangsu 210023, China
Yushun Wang*
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Jiangsu 210023, China
Qikui Du*
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Jiangsu 210023, China
*
*Corresponding author. Email:[email protected] (Q.Hong), [email protected] (Y. S.Wang), [email protected] (Q. K. Du)
*Corresponding author. Email:[email protected] (Q.Hong), [email protected] (Y. S.Wang), [email protected] (Q. K. Du)
*Corresponding author. Email:[email protected] (Q.Hong), [email protected] (Y. S.Wang), [email protected] (Q. K. Du)
Get access

Abstract

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.

Keywords

Generalized fifth-order KdV equationlocal energy-preservingglobal energy-preservingaverage vector fieldFourier pseudospectral

MSC classification

Secondary: 78A48: Composite media; random media
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 5 , October 2017 , pp. 1206 - 1224
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bibi, N., Tirmizi, S. I. A. and Haq, S., Meshless method of lines numerical solution of Kawahara-Type equations, Appl. Math., 2 (2011), pp. 608–618.CrossRefGoogle Scholar
[2] Bridges, T. J., Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), pp. 147–190.CrossRefGoogle Scholar
[3] Bridges, T. J. and Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), pp. 184–193.CrossRefGoogle Scholar
[4] Cai, J. X. and Wang, Y. S., A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schrödinger system, China Phys. B, 22 (2013), 060207.CrossRefGoogle Scholar
[5] Cai, J. X. and Wang, Y. S., Local structure-preserving algorithms for the “good” Boussinesq equation, J. Comput. Phys., 239 (2013), pp. 72–89.CrossRefGoogle Scholar
[6] Cai, J. X., Wang, Y. S. and Liang, H., Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schröinger system, J. Comput. Phys., 239 (2013), pp. 30–50.CrossRefGoogle Scholar
[7] Ceballos, J. C., Sepúlveda, M. and Villagrán, O. P. V., The KdV-Kawahara equation in a bounded domain and some numerical results, Appl. Math. Comput., 190 (2007), pp. 912–936.Google Scholar
[8] Celledoni, E., Grimm, V., McLachlan, R. I., McLaren, D. I., O'Neale, D., Owren, B. and Quispel, G. R. W., Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method, J. Comput. Phys., 231 (2012), pp. 6770–6789.CrossRefGoogle Scholar
[9] Celledoni, E., McLachlan, R. I., McLaren, D. I., Owren, B., Quispel, G. R. W. and Wright, W. M., Energy-preserving Runge-Kutta methods, ESAIM: Math. Model. Numer. Anal., 43 (2009), pp. 645–649.CrossRefGoogle Scholar
[10] Chen, Y., Sun, Y. J. and Tang, Y. F., Energy-preserving numerical methods for Landau-Lifshitz equation, J. Phys. A Math. Theor., 44 (2011), 295207.CrossRefGoogle Scholar
[11] Chen, Y. M., Zhu, H. J. and Song, S. H., Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 181 (2010), pp. 1231–1241.CrossRefGoogle Scholar
[12] Cui, Y. F. and Mao, D. K., Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation, J. Comput. Phys., 227 (2007), pp. 376–399.CrossRefGoogle Scholar
[13] Fei, Z. and Vázquez, L., Two energy conserving numerical schemes for the Sine-Gordon equation, Appl. Math. Comput., 45 (1991), pp. 17–30.Google Scholar
[14] Feng, K. and Qin, M. Z., The Symplectic Methods for Computation of Hamiltonian Systems, Berlin: Springer, (1987), pp. 1–37.Google Scholar
[15] Feng, K. and Qin, M. Z., Symplectic Geometric Algorithms for Hamiltonian Systems, Berlin/Hangzhou: Springer-Verlag/Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, 2003.Google Scholar
[16] Fla, T., A numerical energy conserving method for the DNLS equation, J. Comput. Phys., 101 (1992), pp. 71–79.CrossRefGoogle Scholar
[17] Furihata, D., Finite difference schemes for that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), pp. 181–205.CrossRefGoogle Scholar
[18] 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.CrossRefGoogle Scholar
[19] Hairer, E., Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5 (2010), pp. 73–84.Google Scholar
[20] Hairer, E., Lubich, C. and Wanner, G., Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Diffenential Equations, Berlin: Springer-Verlag, 2006.Google Scholar
[21] Hu, W. P. and Deng, Z. C., Multi-symplectic method for generalized fifth-order KdV equation, China Phys. B, 17 (2008), 3923.Google Scholar
[22] Kaya, D. and Al-Khaled, K., A numerical comparision of a Kawahara equation, Phys. Lett. A, 363 (2007), pp. 433–439.CrossRefGoogle Scholar
[23] Marsden, J., Patrick, G. and Shkoller, S., Mulltisymplectic geometry, variational integrators and nonlinear PDEs, Commun. Math. Phys., 199 (1998), pp. 351–395.CrossRefGoogle Scholar
[24] Matsuo, T. and Furihata, D., Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171 (2001), pp. 425–447.CrossRefGoogle Scholar
[25] McLachlan, R. I., Quispel, G. R. W. and Robidoux, N., Geometric integration using discrete gradients, Philos. Trans. R. Soc. A, 357 (1999), pp. 1021–1046.CrossRefGoogle Scholar
[26] Quispel, G. R. W. and McLaren, D. I., A new class of energy-preserving numerical integration methods, J. Phys. A Math. Theor., 41 (2008), 045206.CrossRefGoogle Scholar
[27] Shen, J. and Tang, T., Spectral and High-order Methods with Applications, Science Press, 2006.Google Scholar
[28] Wang, Y. S., Wang, B. and Qin, M. Z., Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A, 51 (2008), pp. 2115–2136.CrossRefGoogle Scholar
[29] Yuan, J. M., Shen, J. and Wu, J. H., A Dual-Petrov-Galerkin method for the Kawahara-Type equation, J. Sci. Comput., 34 (2008), pp. 48–63.CrossRefGoogle Scholar