An Entropy Stable Scheme for Shallow Water Equations With Source Terms

ZHANG Haijun; FENG Jianhu; CHENG Xiaohan; LI Xue

doi:10.21656/1000-0887.380195

Volume 39 Issue 8

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ZHANG Haijun, FENG Jianhu, CHENG Xiaohan, LI Xue. An Entropy Stable Scheme for Shallow Water Equations With Source Terms[J]. Applied Mathematics and Mechanics, 2018, 39(8): 935-945. doi: 10.21656/1000-0887.380195

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An Entropy Stable Scheme for Shallow Water Equations With Source Terms

doi: 10.21656/1000-0887.380195

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

Funds: The National Natural Science Foundation of China（11601037;11401045;11171043）

Received Date: 2017-07-13
Rev Recd Date: 2017-12-07
Publish Date: 2018-08-15

Abstract

Abstract

An entropy stable scheme was developed for the shallow water equations with source terms, and a 1st-order entropy stable scheme and a high-order entropy conservation scheme were combined with a flux limiter function. The new entropy scheme preserves advantages of both the entropy conservation scheme and the entropy stable scheme, having higher accuracy in the regions of the smooth solutions and capturing shocks accurately while avoiding non-physical phenomena in the regions of the discontinuous solutions, thus achieves high resolution. The new scheme was successfully applied to calculate the classical 1D and 2D problems. The numerical results show that the new scheme does be an ideal method to simulate the shallow water equations with source terms.
- shallow water equations,
- entropy conservation scheme,
- entropy stable scheme,
- high resolution

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References(16)

References

[1]	FJORDHOLM U S, MISHRA S, TADMOR E. Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography[J]. Journal of Computational Physics,2011,230(14): 5587-5609.
[2]	LAX P D. Weak solutions of nonlinear hyperbolic equations and their numerical computation[J]. Communications on Pure and Applied Mathematics,1954,7(1): 159-193.
[3]	LAX P D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves[C]// SIAM Regional Conference Lectures in Applied Mathematics. Vol11. Philadelphia, USA, 1973.
[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]	ROE P L. Entropy conservation schemes forthe Euler equations[R]. Talk at HYP 2006, Lyon, France, 2006.
[6]	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.
[7]	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.
[8]	LIU Y, FENG J, REN J. High resolution, entropy-consistent scheme using flux limiter for hyperbolic systems of conservation laws[J]. Journal of Scientific Computing,2015,64(3): 914-937.
[9]	任炯, 封建湖, 刘友琼, 等. 求解双曲守恒律方程的高分辨率熵相容格式[J]. 计算物理, 2014,31(5): 539-551.(REN Jiong, FENG Jianhu, LIU Youqiong, et al. High resolution entropy consistent schemes for hyperbolic conservation laws[J]. Chinese Journal of Computational Physics,2014,31(5): 539-551.(in Chinese))
[10]	刘友琼, 封建湖, 梁楠, 等. 求解浅水波方程的熵相容格式[J]. 应用数学和力学, 2013,34(12): 1247-1257.(LIU Youqiong, FENG Jianhu, LIANG Nan, et al. An entropy-consistent flux scheme for shallow water equations[J]. Applied Mathematics and Mechanics,2013,34(12): 1247-1257.(in Chinese))
[11]	刘友琼, 封建湖, 任炯, 等. 求解多维Euler方程的二阶旋转混合型格式[J]. 应用数学和力学, 2014,35(5): 542-553.(LIU Youqiong, FENG Jianhu, REN Jiong, et al. A second-order rotated-hybrid scheme for solving multi-dimensional compressible Euler equations[J]. Applied Mathematics and Mechanics,2014,35(5): 542-553.(in Chinese))
[12]	程晓晗, 聂玉峰, 蔡力. 基于WENO重构的熵稳定格式求解浅水方程[J]. 计算物理, 2015,32(5): 523-528.(CHENG Xiaohan, NIE Yufeng, CAI Li. WENO based entropy stable scheme for shallow water equations[J]. Chinese Journal of Computational Physics,2015,32(5): 523-528.(in Chinese))
[13]	程晓晗, 封建湖, 聂玉峰. 求解双曲守恒律方程的WENO型熵相容格式[J]. 爆炸与冲击, 2014,34(4): 501-507.(CHENG Xiaohan, FENG Jianhu, NIE Yufeng. WENO type entropy consistent scheme for hyperbolic conservation laws[J]. Explosion and Shock Waves,2014,34(4): 501-507.(in Chinese))
[14]	郑素佩, 封建湖, 刘彩侠. 高分辨率熵相容算法在二维溃坝问题中的应用[J]. 水动力学研究与进展, 2013,28(5): 545-551.(ZHENG Supei, FENG Jianhu, LIU Caixia. High-resolution entropy consistent algorithm for the two-dimensional dam-break flows[J]. Chinese Journal of Hydrodynamics,2013,28(5): 545-551.(in Chinese))
[15]	GOTTLIEB S, SHU C W. A survey of strong stability preserving high order time discretizations[J]. SIAM Review,2001,43(1): 89-112.
[16]	FJORHOLM U S. Structure preserving finite volume methods for the shallow water equations[D]. Master Thesis. Norway: University of Oslo, 2009.