Second order conformal multi-symplectic method for the damped Korteweg-de Vries equation

doi:10.1088/1674-1056/28/5/050201

Abstract A conformal multi-symplectic method has been proposed for the damped Korteweg-de Vries (DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme, we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations (PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.

Keywords: conformal multi-symplectic method damped Korteweg-de Vries (KdV) equation dissipation preservation

Received: 14 January 2019
Revised: 07 March 2019
Accepted manuscript online:

PACS:	02.60.Lj	(Ordinary and partial differential equations; boundary value problems)
	02.70.Bf	(Finite-difference methods)
	02.60.Cb	(Numerical simulation; solution of equations)

Fund: Project supported by the Program for Innovative Research Team in Science and Technology in Fujian Province University, China, the Quanzhou High Level Talents Support Plan, China (Grant No. 2017ZT012), and the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University, China (Grant No. ZQN-YX502).

Corresponding Authors: Feng Guo
E-mail: hydhgf@163.com

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Feng Guo(郭峰) Second order conformal multi-symplectic method for the damped Korteweg-de Vries equation 2019 Chin. Phys. B 28 050201

[1]	Grimshaw R, Pelinovsky E and Talipova T 2003 Wave Motion 37 351
[2]	Samiran G 2012 EPL 99 36002
[3]	Khan S, Rahman A, Hadi F, Zeb A and Khan M Z 2017 Contrib. Plasma Phys. 57 223
[4]	Tamang J, Sarkar K and Saha A 2018 Physica A 505 18
[5]	Song Z B, Yang X Y, Feng W X, Xi Z H, Li L J and Shi Y R 2018 Chin. Phys. B 27 074501
[6]	Vliegenthart A C 1971 J. Eng. Math. 5 137
[7]	Feng B F and Mitsui T 1998 J. Comput. Appl. Math. 90 95
[8]	Yan J and Shu C W 2002 SIAM J. Numer. Anal. 40 769
[9]	Li J, Ma H P and Sun W W 2000 Numer. Methods Partial Differ. Equations 16 513
[10]	Zhao P F and Qin M Z 2000 J. Phys. A: Math. Gen. 33 3613
[11]	Hong J L and Liu Y 2004 Math. Comput. Modell. 39 1035
[12]	Cui Y F and Mao D K 2007 J. Comput. Phys. 227 376
[13]	Cai J X, Hong Q and Yang B 2017 Chin. Phys. B 26 100202
[14]	Wang H P, Wang Y S and Hu Y Y 2008 Chin. Phys. Lett. 25 2335
[15]	Chen Y M, Song S H and Zhu H J 2012 Appl. Math. Comput. 218 5552
[16]	Marsden J E, Patrick G W and Shkoller S 1998 Commun. Math. Phys. 199 351
[17]	Bridges T J and Reich S 2001 Phys. Lett. A 284 184
[18]	Wang Y S, Wang B and Qin M Z 2008 Sci. China. Ser. A 51 2115
[19]	McLachlan R and Perlmutter M 2001 J. Geom. Phys. 39 276
[20]	McLachlan R I and Quispel G R W 2001 Nonlinearity 14 1689
[21]	Moore B E, Norena L and Schober C M 2013 J. Comput. Phys. 232 214
[22]	Moore B E 2017 J. Comput. Appl. Math. 323 1
[23]	Bhatt A, Floyd D and Moore B E 2016 J. Sci. Comput. 66 1234
[24]	Strang G 1968 SIAM J. Numer. Anal. 5 506
[25]	McLachlan R I and Quispel G R W 2002 Acta Numer. 11 341
[26]	Moore B E 2009 Math. Comput. Simulat. 80 20
[27]	Kong L H, Hong J L and Zhang J J 2010 J. Comput. Phys. 229 4259
[28]	Ma Y P, Kong L H, Hong J L and Cao Y 2011 Comput. Math. Appl. 61 319
[29]	Weng Z F, Zhai S Y and Feng X L 2017 Appl. Math. Modell. 42 462

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Second order conformal multi-symplectic method for the damped Korteweg-de Vries equation

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