The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems
Applied and Computational Mathematics
Volume 4, Issue 2, April 2015, Pages: 39-46
Received: Dec. 22, 2014;
Accepted: Feb. 6, 2015;
Published: Mar. 6, 2015
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Qiong Tang, College of Science, Hunan University of Technology, Zhuzhou, Hunan, P.R. China
Luohua Liua, College of Science, Hunan University of Technology, Zhuzhou, Hunan, P.R. China
Yujun Zheng, Department of mathematics and Computational Science, Hunan University of Science and Engineering, YongZhou, Hunan, P.R. China
Combined with the characteristics of separable Hamiltonian systems and the finite element methods of ordinary differential equations, we prove that the composition of linear, quadratic, cubic finite element methods are symplectic integrator to separable Hamiltonian systems, i.e. the symplectic condition is preserved exactly, but the energy is only approximately conservative after compound. These conclusions are confirmed by our numerical experiments.
The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems, Applied and Computational Mathematics.
Vol. 4, No. 2,
2015, pp. 39-46.
S.Blanes, “High order numerical integrators for differential equations using composition and processing of low order methods,”Appl. Numer. Math., vol.37,pp.289-306,2001.
K. Feng, M. Z. Qin, Symplectic Geometry Algorithm for Hamiltonian systems. ZheJiang Press of Science and Technology, HangZhou, 2003,pp.10-220.
K. Feng, M. Z. Qin, “Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study,” Comput. Phys. Comm., vol.65, pp.173-187 ,1991.
K. Feng, D. L. Wang, “On variation of schemes by Euler,” J. Comp. Math., vol.16, pp.97-106,1998.
C. Kane, J. E. Marsden, M. Ortiz, “Symplectic-Energy-Momentum Preserving Variational Integrators,” J. Math. Phys., vol.40, pp.3353-3371 ,1999.
B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics. Cambridge Universty Press, Cambridge, 2004, pp.16–105..
W. X. Zhong, Z. Yao, “Time Domain FEM and Symplectic Conservation,” Journal of Mechanical Strength, vol.27,pp. 178-183 ,2005.
Q. Tang, C. M. Chen, “Energy conservation and symplectic properties of continuous finite element methods for Hamiltonian systems,”.Appl. Math. and Comp.,vol. 181, pp.1357-1368 ,2006.
Q. Tang, C. M. Chen, L. H. Liu, “Finite element methods for Hamiltonian systems,’ Mathematica Numerica Sinica, vol.31, pp.393-406 ,2009.
C. M. Chen, Y. Q. Huang, High accuracy theory of finite element. Hunan Press of Science and Technology, Changsha, 1995,pp.70-150.
C. M. Chen, Finite element superconvergence construction theory. Hunan Press of Science and Technology, Changsha, 2001,pp.80-158.
K. Feng, Collected Works of Feng Kang. National Defence Industry Press, Beijing, 1995,pp.10-80.
Z. Ge, J. E. Marsden, “Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory.,”Phys. Lett. A,vol. 133, pp.134-139 ,1998.
A. Dullweber, B. Leimkuhler, R.I. McLachlan, “Split-Hamiltonian methods for rigid body molecular dynamics,” J. Chem. Phys., vol.107,pp. 5840- 5851 ,1997.