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

Volume 37 Issue 1

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2016 > 37(1): 1-13

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

Citation:

PDF( 1740 KB)

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

FullText(HTML)

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))

Relative Articles

[1]	MAN Shumin, GAO Qiang, ZHONG Wanxie. A StructurePreserving Algorithm for Hamiltonian Systems With Nonholonomic Constraints[J]. Applied Mathematics and Mechanics, 2020, 41(6): 581-590. doi: 10.21656/1000-0887.400375
[2]	WEN Fen-qiang, DENG Zi-chen, WEI Yi, LI Qing-jun. Dynamic Modelling and Symplectic Solution of Coupled Orbit & Attitude for Solar Sail Towers[J]. Applied Mathematics and Mechanics, 2017, 38(7): 762-768. doi: 10.21656/1000-0887.370321
[3]	LIU You-qiong, FENG Jian-hu, LIANG Nan, REN Jiong. An Entropy-Consistent Flux Scheme for Shallow Water Equations[J]. Applied Mathematics and Mechanics, 2013, 34(12): 1247-1257. doi: 10.3879/j.issn.1000-0887.2013.12.003
[4]	CHEN Guang-hua, CHEN Guang-ming, DAI Zhi-hua. Modified Domain Decomposition Method for Hamilton-Jacobi-Bellman Equations[J]. Applied Mathematics and Mechanics, 2010, 31(12): 1496-1502. doi: 10.3879/j.issn.1000-0887.2010.12.010
[5]	LI Yuan, DUAN Ya-li, GUO Yan, LIU Ru-xun. Numerical Simulation of Laminar Jet-Forced Flow Using a Lattice Boltzmann Method[J]. Applied Mathematics and Mechanics, 2009, 30(4): 417-424.
[6]	LI Wei-hua, LUO En, HUANG Wei-jiang. Unconventional Hamilton-Type Variational Principles for Nonlinear Elastodynamics of Orthogonal Cable-Net Structures[J]. Applied Mathematics and Mechanics, 2007, 28(7): 833-842.
[7]	LIU Yan-hong, ZHANG Hui-ming. Variation Principle of Piezothermoelastic Bodies,Canonical Equation and Homogeneous Equation[J]. Applied Mathematics and Mechanics, 2007, 28(2): 176-182.
[8]	HUANG Wei-jiang, LUO En, SHE Hui. Unconventional Hamilton-Type Variational Principles For Dynamics of Reissner Sandwich Plate[J]. Applied Mathematics and Mechanics, 2006, 27(1): 67-74.
[9]	LUO Xiao-hui, LI Yong-le, LUO Xin. Region-Wise Variational Principles and Generalized Variational Principles on Large Strain for Consolidation Theory[J]. Applied Mathematics and Mechanics, 2005, 26(7): 867-874.
[10]	QING Guang-hui, QIU Jia-jun, LIU Yan-hong. Modified H-R Mixed Variational Principle for Magnetoelectroelastic Bodies and State-Vector Equation[J]. Applied Mathematics and Mechanics, 2005, 26(6): 665-670.
[11]	LUO Zhen-dong, ZHU Jiang, ZENG Qing-cun, XIE Zheng-hui. Mixed Finite Element Methods for the Shallow Water Equations Including Current and Silt Sedimentation (Ⅱ)-The Discrete-Time Case Along Characteristics[J]. Applied Mathematics and Mechanics, 2004, 25(2): 166-180.
[12]	LUO Zhen-dong, ZHU Jiang, ZENG Qing-cun, XIE Zheng-hui. Mixed Finite Element Methods for the Shallow Water Equations Including Current and Silt Sedimentation (Ⅰ)-The Continuous-Time Case[J]. Applied Mathematics and Mechanics, 2004, 25(1): 74-84.
[13]	WANG Guang-hui, WANG Lie-heng. Uzawa Type Algorithm Based on Dual Mixed Variational Formulation[J]. Applied Mathematics and Mechanics, 2002, 23(7): 682-688.
[14]	Luo Shaokai. Relativistic Variation Principles and Equation of Motionfor Variable Mass Controllable Mechanical Systems[J]. Applied Mathematics and Mechanics, 1996, 17(7): 645-653.
[15]	Huang Yan, Huang Reifang. The Variational Principle of Thin Shell in Stability[J]. Applied Mathematics and Mechanics, 1995, 16(12): 1079-1086.
[16]	Deng Zi-chen, Zhong Wan-xie. The Application of the Variational Principle in the Constrained Control System[J]. Applied Mathematics and Mechanics, 1994, 15(6): 489-494.
[17]	Zheng Quan-shui, Yang De-pin, Song Gu-quan. Subspace Variational Principle of Rods and Shells[J]. Applied Mathematics and Mechanics, 1992, 13(11): 977-983.
[18]	Hu Ning, Zhang Ru-qing. A Finite Element Parallel Algorithm of the Parametric Variational Principle for Elastic Contact Problems[J]. Applied Mathematics and Mechanics, 1992, 13(7): 583-591.
[19]	Chien Wei-zhang. Variational Principles and Generalized Variational Principles for Nonlinear Elasticity with Finite Dispiacemant[J]. Applied Mathematics and Mechanics, 1988, 9(1): 1-11.
[20]	Chien Wei-zhang. Variational Principles and Generalized Variational Principles in Hydrodynamics of Viscous Fluids[J]. Applied Mathematics and Mechanics, 1984, 5(3): 305-322.

Supplements(0)

Cited By

Cited by

Periodical cited type(11)

1.	赵珂，陈昌义，席炎炎，黄东威，吴锋，钟万勰. 控制棒下落与流体流动的耦合状态方程及其保辛算法. 应用数学和力学. 2022(09): 935-943 . 本站查看
2.	吴锋，钟万勰. 关于《保辛水波动力学》的一个注记. 应用数学和力学. 2019(01): 1-7 . 本站查看
3.	吴锋，姚征，孙雁，钟万勰. 位移浅水内孤立波. 计算力学学报. 2019(03): 297-303 .
4.	吴锋，钟万勰. 内机械激波——海洋激流的一种解释. 应用数学和力学. 2019(08): 823-839 . 本站查看
5.	吴锋，孙雁，姚征，钟万勰. 椭圆余弦波的位移法分析. 计算机辅助工程. 2018(02): 1-5 .
6.	钟万勰，吴锋，孙雁，姚征. 保辛水波动力学. 应用数学和力学. 2018(08): 855-874 . 本站查看
7.	钟万勰，吴锋，孙雁. 浅水机械激波. 应用数学和力学. 2017(08): 845-852 . 本站查看
8.	文奋强，邓子辰，魏乙，李庆军. 太阳帆塔轨道和姿态耦合动力学建模及辛求解. 应用数学和力学. 2017(07): 762-768 . 本站查看
9.	吴锋，钟万勰. 浅水动边界问题的位移法模拟. 计算机辅助工程. 2016(02): 5-13 .
10.	吴锋，钟万勰. 水波位移法中正定势能的影响. 计算机辅助工程. 2016(06): 1-6 .
11.	张鸿庆，梅建琴. 数学机械化与数学道理化. 应用数学和力学. 2016(07): 665-677 . 本站查看