求解双曲守恒律方程的自适应人工黏性熵稳定格式

任炯; 封建湖; 梁楠; 刘友琼

doi:10.13224/j.cnki.jasp.2014.08.022

求解双曲守恒律方程的自适应人工黏性熵稳定格式

doi: 10.13224/j.cnki.jasp.2014.08.022

长安大学理学院, 西安 710064

基金项目:

国家自然科学基金（11171043）；长安大学中央高校基本科研业务费项目（CHD2102TD015）

详细信息

作者简介:
任炯（1985-），女，山西吕梁人，硕士生，主要从事科学与工程中的高性能计算技术研究。

中图分类号: V411.3;O354;O241.82
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出版历程
- 收稿日期: 2013-05-06
- 刊出日期: 2014-08-28

Adaptive artificial viscosity entropy stable scheme for hyperbolic conservation laws

School of Sciences, Chang'an University, Xi'an 710064, China

摘要

摘要: 采用熵守恒格式求解双曲守恒律方程不满足熵稳定条件.为了达到熵稳定，需要在熵守恒格式的基础上增加黏性机制，使总熵耗散. 鉴于自适应人工黏性在不同的计算区域上其黏性大小的变化特点，在熵守恒通量的基础上添加经过修正的自适应人工黏性通量构造出一种新的结构简单的熵稳定格式. 根据自适应人工黏性的自适应性，并经过简单调节黏性比例系数C之后，该格式可以在不使用限制器的情况下达到整体的高分辨率，表现出：在间断区域有足够的数值耗散保证稳定；在光滑区域，黏性很小，不会影响格式在此的高精度.最后的几个数值算例以熵稳定ERoe格式和高分辨率熵稳定EYee格式的数值结果作为参照，分析说明这种格式的特性.
- 双曲守恒律 /
- 弱局部剩余量 /
- 自适应人工黏性 /
- 熵稳定 /
- 高分辨率
Abstract: The entropy stable conditions may not be met if entropy conservation schemes are used to solve the equations of hyperbolic conservation laws. To achieve entropy stability, the total entropy of a numerical scheme must be dissipative. The frequently-used method is to add a viscosity mechanism to an entropy conservative scheme. By considering the change characteristics of the size of adaptive artificial viscosity in different parts of the computational domain, a new simple entropy stable scheme was proposed by combining the flux of entropy conservation with the corrected flux of adaptive artificial viscosity. According to the adaptivity of adaptive artificial viscosity, this scheme can achieve the overall high resolution without using any limiters but adjusting the viscosity proportional coefficients C. So the computed solution was stable at its discontinuous parts with enough numerical dissipation while maintaining the high order accuracy at smooth parts with very small viscosity. At the end, several numerical examples were performed to analyze the characteristics of the proposed scheme with reference to the numerical results of entropy stable ERoe scheme and high resolution entropy stable EYee scheme.
- hyperbolic conservation laws /
- weak local residual(WLR) /
- adaptive artificial viscosity /
- entropy stable /
- high resolution

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参考文献(16)

[1]	Lax P D.Weak solutions of non-linear hyperbolic equations and their numerical computations[J].Communication on Pure and Applied Mathematics,1954,7(1):159-193.
[2]	Lax P D.Hyperbolic systems of conservation laws and the mathematical theory of shock waves[C]//SIAM Regional Conferences Lectures in Applied Mathematics.Philadelphia:Society for Industrial and Applied Mathematics,1973:1-48.
[3]	Tadmor E.The numerical viscosity of entropy stable schemes for systems of conservation laws:Ⅰ[J].Mathematics of Computation,1987,49(179):91-103.
[4]	Tadmor E,Zhong W.Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity[J].Journal of Hyperbolic Differential Equation, 2006,3(3):529-559.
[5]	Tadmor E,Zhong W.Energy preserving and stable approximations for the two-dimensional shallow water equations[M]//Munthe-Kass H,Owren B.Mathematics and Computation:a Contemporary View.Alesund,Norway:Springer,2008:67-94.
[6]	Roe P L.Entropy conservative schemes for Euler equations[R].Lyon:Eleventh International Conference on Hyperbolic Problems Theory,Numerics,Applications,2006.
[7]	Yee H C.Construction of explicit and implicit symmetric TVD schemes and their applications[J].Chinese Journal of Computational Physics,1987,68(1):151-179.
[8]	罗力,封建湖,唐小娟,等.求解双曲守恒律方程的高分辨率熵稳定格式[J].计算物理,2010,27(5):671-678. LUO Li,FENG Jianhu,TANG Xiaojuan,et al.High resolution entropy stable schemes for hyperbolic conservation laws[J].Journal of Computational Physics,2010,27(5):671-678.(in Chinese)
[9]	Von Neumann J,Richtmyer R.A method for the numerical calculation of hydrodynamic shocks[J].Journal of Applied Physics,1950,21(3):232-237.
[10]	Guermond J L,Pasquetti R.Entropy-based nonlinear viscosity for Fourier approximations of conservation laws[J].Comptes Rendus Mathematique,2008,346(13):801-806.
[11]	Guermond J L,Pasquetti R,Popov B.Entropy viscosity method for nonlinear conservation laws[J].Journal of Computational Physics,2011,230(11):4248-4267.
[12]	Kurganov A,Liu Y.New adaptive artificial viscosity method for hyperbolic systems of conservation laws[J].Journal of Computational Physics,2012,231(24):8114-8132.
[13]	Tadmor E.Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems[J].Acta Numerica,2003,12(1):451-512.
[14]	Ismail F,Roe P L.Afformdable,entropy-consistent Euler flux functions Ⅱ:entropy production at shocks[J].Journal of Computational Physics,2009,228(15):5410-5436.
[15]	Karni S,Kurganov A.Local error analysis for approximate solutions of hyperbolic conservatios laws[J].Advances in Computational Mathematics,2005,22(1):79-99.
[16]	Gottlieb S,Shu C W,Tadmor E.Strong stability-preserving high-order time discretizations methods[J].Society for Industrial and Applied Mathematics Review,2001,43(1):89-112.