The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems

Qiong Tang; Luohua Liua; Yujun Zheng

doi:doi:10.11648/j.acm.20150402.12

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The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems

Qiong Tang, Luohua Liua, Yujun Zheng

Published in Applied and Computational Mathematics (Volume 4, Issue 2)

Received: 22 December 2014 Accepted: 6 February 2015 Published: 6 March 2015

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Abstract

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.

Published in	Applied and Computational Mathematics (Volume 4, Issue 2)
DOI	10.11648/j.acm.20150402.12
Page(s)	39-46
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Separable Hamiltonian Systems, Finite Element Methods, Composition Methods, Symplectic Integrator

References

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[3]	K. Feng, M. Z. Qin, “Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study,” Comput. Phys. Comm., vol.65, pp.173-187 ,1991.
[4]	K. Feng, D. L. Wang, “On variation of schemes by Euler,” J. Comp. Math., vol.16, pp.97-106,1998.
[5]	C. Kane, J. E. Marsden, M. Ortiz, “Symplectic-Energy-Momentum Preserving Variational Integrators,” J. Math. Phys., vol.40, pp.3353-3371 ,1999.
[6]	B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics. Cambridge Universty Press, Cambridge, 2004, pp.16–105..
[7]	W. X. Zhong, Z. Yao, “Time Domain FEM and Symplectic Conservation,” Journal of Mechanical Strength, vol.27,pp. 178-183 ,2005.
[8]	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.
[9]	Q. Tang, C. M. Chen, L. H. Liu, “Finite element methods for Hamiltonian systems,’ Mathematica Numerica Sinica, vol.31, pp.393-406 ,2009.
[10]	C. M. Chen, Y. Q. Huang, High accuracy theory of finite element. Hunan Press of Science and Technology, Changsha, 1995,pp.70-150.
[11]	C. M. Chen, Finite element superconvergence construction theory. Hunan Press of Science and Technology, Changsha, 2001,pp.80-158.
[12]	K. Feng, Collected Works of Feng Kang. National Defence Industry Press, Beijing, 1995,pp.10-80.
[13]	Z. Ge, J. E. Marsden, “Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory.,”Phys. Lett. A,vol. 133, pp.134-139 ,1998.
[14]	A. Dullweber, B. Leimkuhler, R.I. McLachlan, “Split-Hamiltonian methods for rigid body molecular dynamics,” J. Chem. Phys., vol.107,pp. 5840- 5851 ,1997.

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APA Style

Qiong Tang, Luohua Liua, Yujun Zheng. (2015). The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems. Applied and Computational Mathematics, 4(2), 39-46. https://doi.org/10.11648/j.acm.20150402.12

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ACS Style

Qiong Tang; Luohua Liua; Yujun Zheng. The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems. Appl. Comput. Math. 2015, 4(2), 39-46. doi: 10.11648/j.acm.20150402.12

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AMA Style

Qiong Tang, Luohua Liua, Yujun Zheng. The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems. Appl Comput Math. 2015;4(2):39-46. doi: 10.11648/j.acm.20150402.12

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@article{10.11648/j.acm.20150402.12,
  author = {Qiong Tang and Luohua Liua and Yujun Zheng},
  title = {The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {2},
  pages = {39-46},
  doi = {10.11648/j.acm.20150402.12},
  url = {https://doi.org/10.11648/j.acm.20150402.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150402.12},
  abstract = {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.},
 year = {2015}
}

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TY  - JOUR
T1  - The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems
AU  - Qiong Tang
AU  - Luohua Liua
AU  - Yujun Zheng
Y1  - 2015/03/06
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150402.12
DO  - 10.11648/j.acm.20150402.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 39
EP  - 46
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150402.12
AB  - 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.
VL  - 4
IS  - 2
ER  -

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Author Information

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

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APA Style

Qiong Tang, Luohua Liua, Yujun Zheng. (2015). The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems. Applied and Computational Mathematics, 4(2), 39-46. https://doi.org/10.11648/j.acm.20150402.12

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ACS Style

Qiong Tang; Luohua Liua; Yujun Zheng. The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems. Appl. Comput. Math. 2015, 4(2), 39-46. doi: 10.11648/j.acm.20150402.12

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AMA Style

Qiong Tang, Luohua Liua, Yujun Zheng. The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems. Appl Comput Math. 2015;4(2):39-46. doi: 10.11648/j.acm.20150402.12

Copy | Download

@article{10.11648/j.acm.20150402.12,
  author = {Qiong Tang and Luohua Liua and Yujun Zheng},
  title = {The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {2},
  pages = {39-46},
  doi = {10.11648/j.acm.20150402.12},
  url = {https://doi.org/10.11648/j.acm.20150402.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150402.12},
  abstract = {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.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems
AU  - Qiong Tang
AU  - Luohua Liua
AU  - Yujun Zheng
Y1  - 2015/03/06
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150402.12
DO  - 10.11648/j.acm.20150402.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 39
EP  - 46
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150402.12
AB  - 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.
VL  - 4
IS  - 2
ER  -

Copy | Download