Chin. Phys. Lett.  2013, Vol. 30 Issue (3): 030201    DOI: 10.1088/0256-307X/30/3/030201
GENERAL |
A New Multi-Symplectic Integration Method for the Nonlinear Schrödinger Equation
LV Zhong-Quan1,2, WANG Yu-Shun1,3, SONG Yong-Zhong1**
1Jiangsu Key Laboratory for NSLSCS School of Mathematical Science, Nanjing Normal University, Nanjing 210046
2College of Science, Nanjing Forestry University, Nanjing 210037
3Lasg, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029
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LV Zhong-Quan, WANG Yu-Shun, SONG Yong-Zhong 2013 Chin. Phys. Lett. 30 030201
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Abstract

We propose a new multi-symplectic integration method for the nonlinear Schrödinger equation. The new scheme is derived by concatenating spatial discretization of the multi-symplectic Fourier pseudospectral method with temporal discretization of a symplectic Euler scheme and it is semi-explicit in the sense that it does not need to solve the nonlinear algebraic equations at every time step. We verify that the multi-symplectic semi-discretization of the Schrödinger equation with periodic boundary conditions has N semi-discrete multi-symplectic conservation laws. The discretization in time of the semi-discretization leads to N full-discrete multi-symplectic conservation laws. Numerical results are presented to demonstrate the robustness and the stability.
Received: 13 September 2012      Published: 29 March 2013
PACS:  02.60.Cb (Numerical simulation; solution of equations)  
  02.70.Bf (Finite-difference methods)  
  45.10.Na (Geometrical and tensorial methods)  
  45.20.dh (Energy conservation)  
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https://cpl.iphy.ac.cn/10.1088/0256-307X/30/3/030201       OR      https://cpl.iphy.ac.cn/Y2013/V30/I3/030201
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LV Zhong-Quan
WANG Yu-Shun
SONG Yong-Zhong

[1] Asher U M and Mclachlan R I 2004 Appl. Numer. Math. 48 255

[2] Asher U M and Mclachlan R I 2005 J. Sci. Comput. 25 83

[3] Bridges T J 1997 Math. Proc. Cambridge Philos. Soc. 121 147

[4] Bridges T J and Reich S 2001 Phys. Lett. A 284 184

[5] Bridges T J and Reich S 2001 Physica D 152 491

[6] Chen J B and Qin M Z 2001 Electron. Trans. Numer. Anal. 12 193

[7] Chen J B et al 2002 Comput. Math. Appl. 43 1095

[8] Cai J X et al 2009 J. Math. Phys. 50 033510

[9] Cohen D et al 2008 J. Comput. Phys. 227 5492

[10] Feng K and Qin M Z 2010 Symplectic Geometric Algorithms for Hamiltonian systems (Hangzhou: Splinger) (in Chinese)

[11] Hong J L et al 2006 Appl. Numer. Math. 56 814

[12] Hong J L et al 2007 J. Comput. Phys. 226 1968

[13] Hong J L and Kong L H 2010 Commun. Comput. Phys. 7 613

[14] Islas A L et al 2001 J. Comput. Phys. 173 116

[15] Islas A L and Schober C M 2004 J. Comput. Phys. 197 585

[16] Kong L H et al 2006 Chin. J. Comput. Phys. 23 25

[17] Lv Z Q et al 2011 Chin. Phys. Lett. 28 060205

[18] Moore B and Reich S 2003 Numer. Math. 95 625

[19] Qin M Z and Wang Y S 2011 Structure-Preserving Algorithm for PDEs (Hangzhou: Zhejiang Press for Science and Technology) (in Chinese)

[20] Reich S 2000 J. Comput. Phys. 157 473

[21] Wang J 2009 Comput. Phys. Commun. 180 1063

[22] Wang Y S et al 2007 Chin. Phys. Lett. 24 312

[23] Zhao P F and Qin M Z 2000 J. Phys. A: Math. Gen. 33 3613

[24] Zhu H J et al 2011 Adv. Appl. Math. Mech. 3 663
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