求解浅水波方程的熵相容格式

刘友琼; 封建湖; 梁楠; 任炯

doi:10.3879/j.issn.1000-0887.2013.12.003

求解浅水波方程的熵相容格式

doi: 10.3879/j.issn.1000-0887.2013.12.003

长安大学理学院, 西安 710064

基金项目: 国家自然科学基金资助项目（11171043）；中央高校基本科研业务费资助项目（CHD2102TD015）

详细信息

作者简介:
刘友琼（1989—），女，云南人，硕士生(E-mail: youqiongliu@163.com);封建湖（1960—），男，陕西人，教授，博士，博士生导师（通讯作者. E-mail: jhfeng@chd.edu.cn）.

中图分类号: O354;O241.82
计量
- 文章访问数: 1276
- HTML全文浏览量: 61
- PDF下载量: 993
- 被引次数: 0
出版历程
- 收稿日期: 2013-05-12
- 刊出日期: 2013-12-16

An Entropy-Consistent Flux Scheme for Shallow Water Equations

School of Sciences, Chang’an University, Xi’an 710064, P.R.China

Funds: The National Natural Science Foundation of China（11171043）

摘要

摘要: 提出了一种求解浅水波方程组的熵相容格式．在熵稳定通量中添加特征速度差分绝对值的项来抵消解在跨过激波时所产生的熵增，从而实现熵相容．新的数值差分格式具有形式简单、计算效率高、无需添加任何的人工数值粘性的特点．数值算例充分说明了其显著的优点．利用新格式成功地模拟了不同类型溃坝问题的激波、稀疏波传播及溃坝两侧旋涡的形成，是求解浅水波方程组较为理想的方法.
- 数值模拟 /
- 浅水波方程组 /
- 熵相容格式
Abstract: An entropy-consistent flux scheme was developed for the shallow water equations. To offset the entropy increase with cubic order of the shock strength across shock waves, the term of absolute value of the characteristic velocity difference was added into the entropy stable flux, so as to achieve entropy consistency. The new numerical difference scheme featured extreme simplicity, high efficiency, and none additional artificial numerical viscous terms. Numerical experiments of the proposed scheme adequately demonstrated these advantages. The new scheme successfully simulates both the circular shock water wave propagations and the vortices formed on both sides of the breach in different kinds of dam break problems, thus makes a better method to solve the shallow water equations.
- numerical simulation /
- shallow water equations /
- entropy-consistent scheme

HTML全文

参考文献(13)

[1]	Fjordholm U S. Structure preserving finite volume methods for the shallow water equations[D]. Master Thesis. Norway: University of Oslo, 2009.
[2]	Arakawa A, Lamb V R. Computational design of the basic dynamical process of the UCLA general circulation mode[J]. Methods in Computational Physics,1977,17: 173-265.
[3]	LeVeque R J. Balancing source terms and flux gradients in high-resolution Godunov met-hods: the quasi-steady wave-propagation algorithm[J]. Journal of Computational Physics,1998,146（1）: 346-365.
[4]	Tadmor E. The numerical viscosity of entropy stable schemes for systems of conservation laws I[J]. Mathematics of Computation,1987,49(179): 91-103.
[5]	Tadmor E. Entropy stability theory for difference approximation of nonlinear conservation laws and related time-dependent problems[J]. Acta Numerica,2003,12: 451-512.
[6]	Tadmor E, Zhong W. Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity[J]. Journal of Hyperbolic Differential Equations,2006,3(3): 529-559.
[7]	Tadmor E, Zhong W. Energy preserving and stable approximations for the two-dimensional shallow water equations[C]//Munthe-Kaas H, Owren B, eds. Mathematics and Computation, a Contemporary View: The Abel Symposium 2006. The 3rd Abel Symposia . Alesund, Norway. Berlin: Springer, 2008: 67-94.
[8]	Ismail F, Roe P L. Affordable, entropy-consistent Euler flux functions II: entropy production at shocks[J]. Journal of Computational Physics,2009,228(15): 5410-5436.
[9]	Mohammed A N, Ismail F. Study of an entropy-consistent Navier-Stokes flux[J]. International Journal of Computational Fluid Dynamics,2013,27(1): 1-14.
[10]	Fjordholm U, Mishra S, Tadmor E. Energy preserving and energy stable schemes for the shallow water equations[C]//Cucker F, Pinkus A, Todd M J, eds. Foundations of Computational Mathematics, Hong Kong 2008.London Mathematical Society Lecture Note Series(363). London: Cambridge University Press, 2009: 93-139.
[11]	Lax P D. Weak solutions of nonlinear hyperbolic equations and their numerical computations[J]. Communications on Pure and Applied Mathematics,1954,7(1): 159-193.
[12]	Lax P D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves[C]// CBMS-NSF Regional Conference Series in Applied Mathematics,1973,11.
[13]	Fennema R J, Chaudhry M H. Explicit methods for 2-D transient free surface flows[J].Journal of Hydraulic Engineering,1990,116(8): 1013-1034.