# Symplectic properties research for finite element methods of Hamiltonian system

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

### Authors

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

### Abstract

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.

### Keywords

• Hamiltonian systems
• continuous finite element methods
• energy conservative
• wedge product
• symplectic algorithm

•  65P10
•  65L60
•  65L05

### References

• [1] S. D. Bond, B. J. Leimkuhler, Molecular dynamics and the accuracy of numerically computed averages, Acta Numer., 16 (2007), 1–65.

• [2] C. M. Chen, Finite element superconvergence construction theory, Hunan Press of Science and Technology, Changsha (2001)

• [3] C. M. Chen, Y. Q. Huang, High accuracy theory of finite element, Hunan Press of Science and Technology, Changsha (1995)

• [4] K. Feng, Collected Works of Feng Kang, National Defence Industry Press, Beijing (1995)

• [5] K. Feng, M. Z. Qin, Symplectic Geometry Algorithm for Hamiltonian systems , ZheJiang Press of Science and Technology, HangZhou (2004)

• [6] Z. Ge, J. E. Marsden, Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory, Phys. Lett. A, 133 (1988), 134–139.

• [7] O. Gonzalez, J. C. Simo, On the stability of symplectic and energy-momentum algorithms for nonlinear Hamiltonian systems with symmetry, Comp. Meth. Appl. Mech. Engi., 134 (1996), 197–222.

• [8] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag Berlin Heidelberg, Berlin (2006)

• [9] C. Kane, J. E. Marsden, M. Ortiz, Symplectic-Energy-Momentum Preserving Variational Integrators, J. Math. Phys., 40 (1999), 3353–3371.

• [10] F. M. Lasagni , Canonical Runge-Kutta methods , Z. Angew. Math. Phys., 39 (1988), 952–953.

• [11] B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics, Cambridge Universty Press, Cambridge (2004)

• [12] S. Reich, Multisymplectic Runge-Kutta methods for Hamiltonian wave equation, J. Comput. Phys., 157 (2000), 473–499.

• [13] R. D. Ruth, A canonical intergration technique, IEEE Trans. Nucl. Sci., 30 (1983), 2669–2671.

• [14] J. M. Sanz-Serna, Runge-Kutta Schemes for Hamiltonian Systems, BIT, 28 (1988), 877–883.

• [15] J. M. Sanz-Serna, M. P. Calvo , Numerical Hamiltonian Problems, Chapman & Hall, London (1994)

• [16] A. M. Stuart, A. R. Humphries , Dynamical Systems and Numerical Analysis, Cambridge university press, Cambridge (1998)

• [17] Y. B. Suris , The canonicity of mappings generated by Runge-Kutta type methods when integrating the systems $x''=-\frac{\partial u}{\partial x}$, U.S.S.R. Comput. Math. and Math. Phys., 29 (1989), 138–144.

• [18] Q. Tang, C.-M. Chen, L.-H. Liu, Energy conservation and symplectic properties of continuous finite element methods for Hamiltonian systems, Appl. Math. Comput., 181 (2006), 1357–1368.