一种健壮的低耗散通量分裂格式

胡立军; 袁礼; 翟健

doi:10.21656/1000-0887.390132

一种健壮的低耗散通量分裂格式

doi: 10.21656/1000-0887.390132

¹衡阳师范学院数学与统计学院, 湖南衡阳 421002;²中国科学院数学与系统科学研究院, 北京 100190;³曙光信息产业(北京)有限公司, 北京 100193

详细信息

作者简介:
胡立军（1985—），男，博士(通讯作者. E-mail: hulijun@lsec.cc.ac.cn).

中图分类号: O354;O241.82
计量
- 文章访问数: 1113
- HTML全文浏览量: 213
- PDF下载量: 489
- 被引次数: 0
出版历程
- 收稿日期: 2018-04-25
- 修回日期: 2018-06-13
- 刊出日期: 2019-02-01

A Robust and Low-Dissipation Flux Splitting Scheme

¹College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan 421002, P.R.China;²Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R.China;³Dawning Information Industry Co., Ltd., Beijing 100193, P.R.China

摘要

摘要: 随着计算流体力学的快速发展,设计精确、高效并且健壮的数值格式变得尤为重要.通过对3种流行的通量分裂方法(AUSM、Zha-Bilgen和Toro-Vázquez)的对流通量和压力通量进行特征分析,构造了一种简单、低耗散并且健壮的通量分裂格式(命名为R-ZB格式).采用Zha-Bilgen分裂方法将Euler方程的通量分裂成对流通量和压力通量,其中对流通量采用迎风方法来计算,压力通量采用低耗散的HLL格式来计算,从而克服了原始的HLL格式不能精确分辨接触间断的缺点.数值实验表明,该文给出的R-ZB格式不仅保留了原始Zha-Bilgen格式简单高效、能够精确分辨接触间断等优点,而且具有更好的健壮性,在计算二维问题时不会出现数值激波不稳定现象.
- Euler方程 /
- Zha-Bilgen分裂 /
- HLL /
- R-ZB /
- 低耗散 /
- 数值激波不稳定性
Abstract: With the rapid development of computational fluid dynamics, it is particularly important to design accurate, efficient and robust numerical schemes. Through the characteristics analyses of 3 popular flux splitting methods (AUSM, Zha-Bilgen and Toro-Vázquez), a simple, low-dissipation and robust flux splitting scheme (named as R-ZB) was constructed. The flux of Euler equations was split into a convection flux and a pressure flux with the Zha-Bilgen splitting procedure. The convection flux was computed with a simple upwinding scheme, and the pressure flux was evaluated with a low-dissipation HLL scheme to overcome the flaw of failing to capture contact discontinuities. Numerical experiments show that, the proposed R-ZB scheme not only retains the merits of the original Zha-Bilgen scheme, such as simpleness, efficiency and capturing contact discontinuities accurately, etc., but also has better robustness, which eliminates the numerical shock instabilities in the calculation of 2D problems.
- Euler equations /
- Zha-Bilgen splitting /
- HLL /
- R-ZB /
- low dissipation /
- numerical shock instability

HTML全文

参考文献(34)

[1]	LIOU M S. Open issues in numerical fluxes: proposed resolutions[C]// 〖STBX〗20th AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences,2011. DOI: 10.2514/6.2011-3055.
[2]	QU F, YAN C, YU J, et al. A new flux splitting scheme for the Euler equations[J]. Computers & Fluids,2014,102: 203-214.
[3]	ROE P L. Approximate Riemann solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics,1981,43(2): 357-372.
[4]	HARTEN A, LAX P D, VAN LEER B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[J]. SIAM Review,1983,25(1): 35-61.
[5]	TORO E F, SPRUCE M, SPEARES W. Restoration of the contact surface in the HLL-Riemann solver[J]. Shock Waves,1994,4(1): 25-34.
[6]	TORO E F. Riemann Solvers and Numerical Methods for Fluid Dynamics [M]. 3rd ed. Berlin: Springer, 1999.
[7]	KIM S D, LEE B J, LEE H J, et al. Robust HLLC Riemann solver with weighted average flux scheme for strong shock[J]. Journal of Computational Physics,2009,228(20): 7634-7642.
[8]	STEGER J L, WARMING R F. Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods[J]. Journal of Computational Physics,1981,40(2): 263-293.
[9]	VAN LEER B. Flux vector splitting for the Euler equations[C]// 〖STBX〗8th International Conference on Numerical Methods in Fluid Dynamics . Berlin, Heidelberg: Springer-Verlag, 1982: 507-512.
[10]	ANDERSON W K, THOMAS J L, VAN LEER B. Comparison of finite volume flux vector splittings for the Euler equations[J]. AIAA Journal,1986,24(9): 1453-1460.
[11]	TORO E F, VZQUEZ-CENDN M E. Flux splitting schemes for the Euler equations[J]. Computers & Fluids,2012,70: 1-12.
[12]	LIOU M S, STEFFEN JR C J. A new flux splitting scheme[J]. Journal of Computational Physics,1993,107(1): 23-39.
[13]	LIOU M S. A sequel to AUSM: AUSM+[J]. Journal of Computational Physics,1996,129(2): 364-382.
[14]	LIOU M S. Recent progress and applications of AUSM+[C]//16th International Conference on Numerical Methods in Fluid Dynamics . Berlin, Germany, 1998.
[15]	LIOU M S. A sequel to AUSM, part II: AUSM+-up for all speeds[J]. Journal of Computational Physics,2006,214(1): 137-170.
[16]	ZHA G C, BILGEN E. Numerical solution of Euler equations by using a new flux vector splitting scheme[J]. International Journal for Numerical Methods in Fluids,1993,17(2): 115-144.
[17]	ZHA G C, SHEN Y, WANG B. An improved low diffusion E-CUSP upwind scheme[J]. Computers & Fluids,2011,48(1): 214-220.
[18]	KAPEN P T, TCHUEN G. An extension of the TV-HLL scheme for multi-dimensional compressible flows[J]. International Journal of Computational Fluid Dynamics,2015,29(3/5): 303-312.
[19]	TORO E F, CASTRO C E, LEE B J. A novel numerical flux for the 3D Euler equations with general equation of state[J]. Journal of Computational Physics,2015,303: 80-94.
[20]	XIE W, LI H, TIAN Z, et al. A low diffusion flux splitting method for inviscid compressible flows[J]. Computers & Fluids,2015,112: 83-93.
[21]	SHIMA E, KITAMURA K. Parameter-free simple low-dissipation AUSM-family scheme for all speeds[J]. AIAA Journal,2011,49(8): 1693-1709.
[22]	MANDAL J C, PANWAR V. Robust HLL-type Riemann solver capable of resolving contact discontinuity[J]. Computers & Fluids,2012,63: 148-164.
[23]	QUIRK J J. A contribution to the great Riemann solver debate[J]. International Journal for Numerical Methods in Fluids,1994,18(6): 555-574.
[24]	HU L J, YUAN L. A robust hybrid HLLC-FORCE scheme for curing numerical shock instability[J]. Applied Mechanics and Materials,2014,577: 749-753.
[25]	WU H, SHEN L J, SHEN Z J. A hybrid numerical method to cure numerical shock instability[J]. Communications in Computational Physics,2010,8: 1264-1271.
[26]	REN Y X. A robust shock-capturing scheme based on rotated Riemann solvers[J]. Computers & Fluids,2003,32(10): 1379-1403.
[27]	胡立军, 袁礼. 一种治愈强激波数值不稳定性的混合方法[J]. 应用数学和力学, 2015,36(5): 482-493.(HU Lijun, YUAN Li. Analysis of numerical shock instability and a hybrid curing method[J]. Applied Mathematics and Mechanics,2015,36(5): 482-493.(in Chinese))
[28]	SUN M, TAKAYAMA K. An artificially upstream flux vector splitting scheme for the Euler equations[J]. Journal of Computational Physics,2003,189(1): 305-329.
[29]	GOTTLIEB S. On high order strong stability preserving Runge-Kutta and multi step time discretizations[J]. Journal of Scientific Computing,2005,25(1/2): 105-128.
[30]	LI B, YUAN L. Convergence issues in using high-resolution schemes and lower-upper symmetric Gauss-Seidel method for steady shock-induced combustion problems[J]. International Journal for Numerical Methods in Fluid,2013,71(11): 1422-1437.
[31]	JIANG G S, SHU W C. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics,1996,126(1): 202-228.
[32]	WOODWARD P, COLELLA P. The numerical simulation of two-dimensional fluid flow with strong shocks[J]. Journal of Computational Physics,1984,54(1): 115-173.
[33]	KITAMURA K, ROE P L, ISMAIL F. Evaluation of Euler fluxes for hypersonic flow computations[J]. AIAA Journal,2009,47(1): 44-53.
[34]	LAX P D, LIU X D. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes[J]. SIAM Journal on Scientific Computing,1998,19(2): 319-340.