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Chin. Phys. B, 2015, Vol. 24(5): 050205    DOI: 10.1088/1674-1056/24/5/050205
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A local energy-preserving scheme for Klein–Gordon–Schrödinger equations

Cai Jia-Xianga b, Wang Jia-Lina, Wang Yu-Shuna
a Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China;
b School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
Abstract  A local energy conservation law is proposed for the Klein–Gordon–Schrödinger equations, which is held in any local time–space region. The local property is independent of the boundary condition and more essential than the global energy conservation law. To develop a numerical method preserving the intrinsic properties as much as possible, we propose a local energy-preserving (LEP) scheme for the equations. The merit of the proposed scheme is that the local energy conservation law can hold exactly in any time–space region. With the periodic boundary conditions, the scheme also possesses the discrete change and global energy conservation laws. A nonlinear analysis shows that the LEP scheme converges to the exact solutions with order O(τ2+h2). The theoretical properties are verified by numerical experiments.
Keywords:  Klein–Gordon–Schrödinger equations      energy conservation law      local structure      convergence analysis  
Received:  28 November 2014      Revised:  09 December 2014      Published:  05 May 2015
PACS:  02.60.Cb (Numerical simulation; solution of equations)  
  02.70.Bf (Finite-difference methods)  
  02.60.Lj (Ordinary and partial differential equations; boundary value problems)  
Fund: Project of Graduate Education Innovation of Jiangsu Province, China (Grant No. CXLX13_366).
Corresponding Authors:  Wang Yu-Shun     E-mail:  thomasjeer@sohu.com, wangyushun@njnu.edu.cn
About author:  02.60.Cb; 02.70.Bf; 02.60.Lj

Cite this article: 

Cai Jia-Xiang, Wang Jia-Lin, Wang Yu-Shun A local energy-preserving scheme for Klein–Gordon–Schrödinger equations 2015 Chin. Phys. B 24 050205

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