浅水波方程熵稳定格式的保平衡性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (2): 491-504.

浅水波方程熵稳定格式的保平衡性

建芒芒(),郑素佩^*(),封建湖(),翟梦情()

长安大学理学院西安 710064

收稿日期:2022-07-25 修回日期:2022-10-17 出版日期:2023-04-26 发布日期:2023-04-17
通讯作者: 郑素佩, E-mail:zsp2008@chd.edu.cn
作者简介:建芒芒, E-mail:j3506687033@163.com|封建湖,E-mail:jhfeng@chd.edu.cn|翟梦情, E-mail:zhaimengqing2016@163.com
基金资助:
国家自然科学基金(11971075);国家自然科学基金(11901057)

Well-Balanced Preserving of Entropy Stable Schemes for Shallow Water Equations

Jian Mangmang(),Zheng Supei(),Feng Jianhu(),Zhai Mengqing()

College of Science, Chang'an University, Xi'an 710064

Received:2022-07-25 Revised:2022-10-17 Online:2023-04-26 Published:2023-04-17
Supported by:
NSFC(11971075);NSFC(11901057)

摘要/Abstract

摘要：

保平衡性是浅水波方程的一个重要性质, 满足保平衡性的格式理论上能够准确捕捉稳态的微小扰动. 针对带有底部地形源项的浅水波方程设计恰当的数值耗散算子并选择合适的源项离散方式以精确平衡非零通量与源项, 构造了一类保平衡的高阶熵稳定格式, 提出了高阶熵守恒格式以及熵稳定格式的保平衡性定理并给予理论证明. 利用新格式计算几个典型的数值算例, 数值结果表明新格式能够很好地处理稳态解的小扰动问题.

关键词: 保平衡性, 稳态的小扰动问题, 熵稳定格式, 浅水波方程

Abstract:

Preserving well-balanced property is an important property for shallow water matrixs. The schemes with this property can capture small perturbations of steady state accurately in theory. For the shallow water matrixs with source terms, the first step is to construct an appropriate numerical dissipative operator and select a special discretization of source terms to accurately balance the non-zero flux and source terms, which is to obtain a class of high order well-balanced entropy stable schemes to keep balance. The new idea is to put forward the well-balanced preserving theorems for the high-order entropy conservative schemes and for the high-order entropy stable schemes. The detail proof process is also clearly given. Finally, several typical numerical examples show that new schemes can well deal with the small perturbation problems of steady-state solutions.

Key words: Well-balanced, Steady-state problems with small perturbations, Entropy stable schemes, Shallow water matrixs

中图分类号:

O354

建芒芒, 郑素佩, 封建湖, 翟梦情. 浅水波方程熵稳定格式的保平衡性[J]. 数学物理学报, 2023, 43(2): 491-504.

Jian Mangmang, Zheng Supei, Feng Jianhu, Zhai Mengqing. Well-Balanced Preserving of Entropy Stable Schemes for Shallow Water Equations[J]. Acta mathematica scientia,Series A, 2023, 43(2): 491-504.

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参考文献 20

[1]	Lin G F, Lai J S, Guo W D. Performance of high-resolution TVD schemes for $1$D dam-break simulations. J Chin Inst Eng, 2005, 28(5): 771-782 doi: 10.1080/02533839.2005.9671047
[2]	Xing Y L, Shu C W. High order finite difference WENO schemes with the exact conservation property for the shallow water matrixs. J Comput Phys, 2005, 208(1): 206-227 doi: 10.1016/j.jcp.2005.02.006
[3]	Kesserwani G, Liang Q H. A conservative high-order discontinuous Galerkin method for the shallow water matrixs with arbitrary topography. Int J Numer Meth Eng, 2011, 86(1): 47-69 doi: 10.1002/nme.v86.1
[4]	郑素佩, 李霄, 赵青宇, 等. 求解二维浅水波方程的旋转混合格式. 应用数学和力学, 2022, 43(2): 176-186
	Zheng S P, Li X, Zhao Q Y, et al. A rotated mixed scheme for solving 2D shallow water matrixs. Appl Math Mech, 2022, 43(2): 176-186
[5]	Kurganov A, Levy D. Central-upwind schemes for the Saint-Venant system. Math Model Anal, 2002, 36(3): 397-425
[6]	Liu X, Albright J, Epshteyn Y, et al. Well-balanced positivity preserving central-upwind scheme with a novel wet/dry reconstruction on triangular grids for the Saint-Venant system. J Comput Phys, 2018, 374: 213-236 doi: 10.1016/j.jcp.2018.07.038
[7]	Kurganov A, Noelle S, Petrov A G. Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi matrixs. SIAM J Sci Comput, 2001, 23(3): 707-740 doi: 10.1137/S1064827500373413
[8]	董建. 中心格式在 Saint-Venant 方程组上的应用研究. 数学物理学报, 2019, 39A(2): 372-385
	Dong J. Research on the application of central scheme in Saint-Venant system. Acta Math Sci, 2019, 39A(2): 372-385
[9]	Tadmor E. The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math Comput, 1987, 49(179): 91-103
[10]	Fjordholm U S, Mishra S, Tadmor E. Well-balanced and energy stable schemes for the shallow water matrixs with discontinuous topography. J Comput Phys, 2011, 230(14): 5587-5609 doi: 10.1016/j.jcp.2011.03.042
[11]	张海军, 封建湖, 程晓晗, 等. 带源项浅水波方程的高分辨率熵稳定格式. 应用数学和力学, 2018, 39(8): 935-945
	Zhang H J, Feng J H, Cheng X H, et al. An entropy stable scheme for shallow water matrixs with source terms. Appl Math Mech, 2018, 39(8): 935-945
[12]	Fjordholm U S, Mishra S, Tadmor E. Arbitrarily high-order accurate entropy stable essentially non-oscillatory schemes for systems of conservation laws. SIAM J Numer Anal, 2012, 50(2): 544-573 doi: 10.1137/110836961
[13]	郑素佩, 建芒芒, 封建湖, 等. 保号WENO-AO型中心迎风格式. 计算物理, http://kns.cnki.net/kcms/detail/11.2011.O4.20220322.1639.008.html
	Zheng S P, Jian M M, Feng J H, et al. Sign preserving WENO-AO-type central upwind schemes. Chinese J Comput Phys, http://kns.cnki.net/kcms/detail/11.2011.O4.20220322.1639.008.html
[14]	Lax P D. Weak solutions of nonlinear hyperbolic matrixs and their numerical computation. Commun Pur Appl Math, 1954, 7(1): 159-193 doi: 10.1002/(ISSN)1097-0312
[15]	Fjordholm U S, Ray D. A sign preserving WENO reconstruction method. J Sci Comput, 2016, 68(1): 42-63 doi: 10.1007/s10915-015-0128-y
[16]	Shu C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer, 2020, 29: 701-762 doi: 10.1017/S0962492920000057
[17]	郑素佩, 徐霞, 封建湖, 等. 高阶保号熵稳定格式. 数学物理学报, 2021, 41A(5): 1296-1310
	Zheng S P, Xu X, Feng J H, et al. High order sign preserving stable schemes. Acta Math Sci, 2021, 41A(5): 1296-1310
[18]	徐霞. 求解双曲守恒律方程的任意阶保号熵稳定格式研究. 西安:长安大学, 2021
	Xu X. The Research of Arbitrary High-Order Sign Prserving Entropy Stable Schemes for Hyperbolic Conservation Laws. Xi'an: Chang'an University, 2021
[19]	Leveque R J. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J Comput Phys, 1998, 146(1): 346-365 doi: 10.1006/jcph.1998.6058
[20]	罗一鸣, 李订芳, 刘敏, 等. 一种维持 Saint-Venant 方程组移动稳态解的中心格式. 数学物理学报, 2022, 42A(3): 891-903
	Luo Y M, Li D F, Liu M, et al. Moving-water equilibria preserving central scheme for the Saint-Venant system. Acta Math Sci, 2022, 42A(3): 891-903