The Constrained Hamilton Variational Principle for Shallow Water Problems and the Zu-Type Symplectic Algorithm

WU Feng; ZHONG Wan-xie

doi:10.3879/j.issn.1000-0887.2016.01.001

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WU Feng, ZHONG Wan-xie. The Constrained Hamilton Variational Principle for Shallow Water Problems and the Zu-Type Symplectic Algorithm[J]. Applied Mathematics and Mechanics, 2016, 37(1): 1-13. doi: 10.3879/j.issn.1000-0887.2016.01.001

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The Constrained Hamilton Variational Principle for Shallow Water Problems and the Zu-Type Symplectic Algorithm

doi: 10.3879/j.issn.1000-0887.2016.01.001

State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology), Dalian, Liaoning 116023, P.R.China

Funds: The National Natural Science Foundation of China(General Program)（11472067）

Received Date: 2015-09-30
Rev Recd Date: 2015-12-01
Publish Date: 2016-01-16

Abstract

Abstract

The shallow water problems were addressed. With the incompressible condition as the constraint, a constrained Hamilton variational principle was proposed for the shallow water problems. Based on the constrained Hamilton variational principle, the corresponding shallow water equations based on the displacement and pressure (SWE-DP) were developed. A hybrid numerical method combining the finite element method for the spatial discretization and the Zu-type symplectic method for the time integration was proposed to solve the SWE-DP. The correctness of the proposed SWE-DP is verified through the numerical comparisons of the present results with those from 2 sets of existing shallow water equations. The feasibility of the hybrid numerical method proposed for the SWE-DP is also proved through the numerical experiments. Moreover, the numerical experiments demonstrate the excellent performance of the Zu-type method for the simulation of the long time evolution of the shallow water motion.
- shallow water equation,
- constrained Hamilton variational principle,
- Zu-type method

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References(14)

References

[1]	兰姆 H. 理论流体动力学[M]. 游镇雄, 牛家玉, 译. 北京: 科学出版社, 1990.(Lamb H.Hydrodynamics[M]. YOU Zhen-xiong, NIU Jia-yu, transl. Beijing: Science Press, 1990.(Chinese version))
[2]	Stoker J J.Water Waves: The Mathematical Theory With Applications [M]. New York: Interscience Publishers Ltd, 1957.
[3]	Vreugdenhil C B.Numerical Methods for Shallow-Water Flow [M]. Netherlands: Springer, 1994.
[4]	Kernkamp H W J, Van Dam A, Stelling G S, de Goede E D. Efficient scheme for the shallow water equations on unstructured grids with application to the continental shelf[J].Ocean Dynamics,2011,61(8): 1175-1188.
[5]	LIU Hai-fei, WANG Hong-da, LIU Shu, HU Chang-wei, DING Yu, ZHANG Jie. Lattice Boltzmann method for the Saint-Venant equations[J].Journal of Hydrology,2015,524: 411-416.
[6]	Chalfen M, Niemiec A. Analytical and numerical-solution of Saint-Venant equations[J].Journal of Hydrology,1986,86(1/2): 1-13.
[7]	Remoissenet M.Waves Called Solitons: Concepts and Experiments [M]. Berlin: Springer, 1996.
[8]	钟万勰. 应用力学的辛数学方法[M]. 北京: 高等教育出版社, 2006.(ZHONG Wang-xie.Symplectic Solution Methodology in Applied Mechanics [M]. Beijing: Higher Education Press, 2006.(in Chinese))
[9]	FENG Kang, QIN Meng-zhao.Symplectic Geometric Algorithms for Hamiltonian Systems [M]. Heidelberg, Berlin: Springer, 2010.
[10]	钟万勰, 陈晓辉. 浅水波的位移法求解[J]. 水动力学研究与进展(A辑), 2006,21(4): 486-493.(ZHONG Wan-xie, CHEN Xiao-hui. Solving shallow water waves with the displacement method[J].Journal of Hydrodynamics(Ser A),2006,21(4): 486-493.(in Chinese))
[11]	钟万勰, 姚征. 位移法浅水孤立波[J]. 大连理工大学学报, 2006,46(1): 151-156.(ZHONG Wan-xie, YAO Zheng. Shallow water solitary waves based on displacement method[J].Journal of Dalian University of Technology,2006,46(1): 151-156.(in Chinese))
[12]	钟万勰, 高强. 约束动力系统的分析结构力学积分[J]. 动力学与控制学报, 2006,4(3): 193-200.(ZHONG Wan-xie, GAO Qiang. Integration of constrained dynamical system via analytical structural mechanics[J].Journal of Dynamics and Control,2006，〖STHZ〗4(3): 193-200.(in Chinese))
[13]	钟万勰, 高强, 彭海军. 经典力学辛讲[M]. 大连: 大连理工大学出版社, 2013.(ZHONG Wan-xie, GAO Qiang, PENG Hai-jun.Classical Mechanics—Its Symplectic Description[M]. Dalian: Dalian University of Technology Press, 2013.(in Chinese))
[14]	吴锋, 钟万勰. 基于祖冲之类方法具有保辛性[J]. 计算力学学报, 2015,32(4): 447-450.(WU Feng, ZHONG Wan-xie. The Zu-type method is symplectic[J].Chinese Journal of Computational Mechanics,2015,32(4): 447-450.(in Chinese))