Symplectic properties research for finite element methods of Hamiltonian system

Volume 18, Issue 3, pp 314--327 Publication Date: June 20, 2018       Article History


Qiong Tang - Department of Information and Mathematics Science, Hunan University of Technology, Hunan, ZhuZhou, 412008, P. R. China Yangfan Liu - College of materials science and engineering, Xiang Tan University, XiangTan 411105, Hunan, P. R. China Yujun Zheng - Department of mathematics and Computational Science, Hunan University of Science and Engineering, YongZhou 425100, Hunan, P. R. China Hongping Cao - College of Management, Shanghai University of Engineering Science, Shanghai, 201620, P. R. China


In this paper, we first apply properties of the wedge product and continuous finite element methods to prove that the linear, quadratic element methods are symplectic algorithms to the linear Hamiltonian systems, i.e., the symplectic condition \(dp_{j+1}\wedge dq_{j+1}=dp_{j}\wedge dq_{j}\) is preserved exactly and the linear element method is an approximately symplectic integrator to nonlinear Hamiltonian systems, i.e., \(dp_{j+1}\wedge dq_{j+1}=dp_{j}\wedge dq_{j}+O(h^2)\), as well as energy conservative.