浅水问题的约束Hamilton变分原理及祖冲之类保辛算法

吴锋; 钟万勰

doi:10.3879/j.issn.1000-0887.2016.01.001

浅水问题的约束Hamilton变分原理及祖冲之类保辛算法

doi: 10.3879/j.issn.1000-0887.2016.01.001

吴锋,
钟万勰

工业装备结构分析国家重点实验室(大连理工大学),辽宁大连 116023

基金项目: 国家自然科学基金（面上项目）（11472067）

详细信息

作者简介:
吴锋（1985—），男，江苏靖江人，博士（通讯作者. E-mail: wufeng_chn@163.com）;钟万勰（1934—），男，浙江德清人，教授，院士（E-mail: zwoffice@dlut.edu.cn）.

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出版历程
- 收稿日期: 2015-09-30
- 修回日期: 2015-12-01
- 刊出日期: 2016-01-16

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

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）

摘要

摘要: 针对浅水流问题，将不可压缩条件作为约束处理，提出一种约束Hamilton变分原理，并利用该变分原理，推出一种基于位移和压强的浅水方程（SWE-DP）.针对SWE-DP，构造了一种结合有限元和祖冲之类算法的混合数值方法.通过数值算例，将SWE-DP与两个现有的浅水方程进行了数值比较，从而验证了SWE-DP的可靠性，并验证了针对SWE-DP构造的数值算法的正确性.此外，数值算例还显示出祖冲之类算法在对浅水波进行长时间仿真时，具有很好的表现.
- 浅水方程 /
- 约束Hamilton变分原理 /
- 祖冲之类算法
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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参考文献(14)

[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))
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[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.
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[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))
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浅水问题的约束Hamilton变分原理及祖冲之类保辛算法

doi: 10.3879/j.issn.1000-0887.2016.01.001

作者简介:
吴锋（1985—），男，江苏靖江人，博士（通讯作者. E-mail: wufeng_chn@163.com）;钟万勰（1934—），男，浙江德清人，教授，院士（E-mail: zwoffice@dlut.edu.cn）.

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

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留言板

浅水问题的约束Hamilton变分原理及祖冲之类保辛算法

doi: 10.3879/j.issn.1000-0887.2016.01.001

作者简介: 吴锋（1985—），男，江苏靖江人，博士（通讯作者. E-mail: wufeng_chn@163.com）;钟万勰（1934—），男，浙江德清人，教授，院士（E-mail: zwoffice@dlut.edu.cn）.

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出版历程

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

计量

出版历程

目录

作者简介:
吴锋（1985—），男，江苏靖江人，博士（通讯作者. E-mail: wufeng_chn@163.com）;钟万勰（1934—），男，浙江德清人，教授，院士（E-mail: zwoffice@dlut.edu.cn）.