高效率的特征型紧致WENO混合格式

骆信; 吴颂平

doi:10.13700/j.bh.1001-5965.2019.0573

高效率的特征型紧致WENO混合格式

doi: 10.13700/j.bh.1001-5965.2019.0573

骆信,
吴颂平^,

北京航空航天大学航空科学与工程学院, 北京 100083

基金项目:

国家自然科学基金 91530325

详细信息

作者简介:
骆信男, 博士研究生。主要研究方向:计算流体力学

吴颂平男, 博士, 教授, 博士生导师。主要研究方向:计算流体力学、有限元方法

通讯作者:
吴颂平, E-mail: wusping825@163.com

中图分类号: V211.3
计量
- 文章访问数: 510
- HTML全文浏览量: 62
- PDF下载量: 218
- 被引次数: 0
出版历程
- 收稿日期: 2019-11-10
- 录用日期: 2020-02-02
- 网络出版日期: 2020-07-20

An efficient characteristic-wise hybrid compact-WENO scheme

LUO Xin,
WU Songping^,

School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China

Funds:

National Natural Science Foundation of China 91530325

More Information

Corresponding author: WU Songping, E-mail: wusping825@163.com

摘要

摘要:
特征型紧致加权基本无振荡（WENO）混合格式HCW-R结合了迎风紧致格式CS5-P和WENO格式，具有十分优异的分辨率特性。但在求解多维方程组时，HCW-R格式需要求解块状三对角方程组，因而计算代价十分高昂。采用迎风紧致格式CS5-F代替CS5-P，构造了一个新的特征型紧致WENO混合格式HCW-E。由于HCW-E的特殊形式，其可沿迎风方向、由边界处向内推进求解，避免了处理三对角或块状三对角方程组，从而其计算代价与显式格式无异。虽然就分辨率而言，HCW-E稍逊于HCW-R，但前者的计算效率要显著高于后者。因此，当花费相同的计算代价，HCW-E格式可以获得更好的数值结果。一系列求解Euler方程组的数值试验验证了HCW-E的高分辨率特性和相比HCW-R更高的计算效率。HCW-E格式的效率优势在求解高维问题时更为明显。
- 紧致格式 /
- 加权基本无振荡(WENO)格式 /
- 混合格式 /
- 高分辨率 /
- 激波捕捉
Abstract:
The characteristic-wise hybrid compact-Weighted Essentially Non-Oscillatory (WENO) scheme HCW-R combines the upwind compact scheme CS5-P with the WENO scheme, achieving an excellent resolution property. However, it needs to deal with the block-tridiagonal systems of linear equations when applied to solve multi-dimensional equations. Therefore, the scheme is computationally expensive. In this paper, we construct the new characteristic-wise hybrid compact-WENO scheme HCW-E, where the upwind compact scheme CS5-F is used to replace CS5-P. Due to the special form of HCW-E, the new hybrid scheme can be solved in an advancing manner, which is along the upwind direction and from the boundary inward. In this way, the tridiagonal or block-tridiagonal systems of linear equations are avoided. As a result, the computational expense of the compact-type scheme is equivalent to that of the explicit scheme. Although the resolution of HCW-E is slightly lower than that of HCW-R, the computational efficiency of the former is significantly higher than that of the latter. Therefore, the new scheme can get better numerical results at the same cost as the previous scheme. A series of numerical experiments for solving Euler equations show that HCW-E has an excellent resolution property and is much more efficient than HCW-R. The superiority of the new scheme in computational efficiency is more significant in solving higher-dimensional problems.
- compact scheme /
- Weighted Essentially Non-Oscillatory (WENO) scheme /
- hybrid scheme /
- high resolution /
- shock capture

HTML全文

图 1 Lax问题的密度分布曲线

Figure 1. Density distribution curves of Lax problem

下载: 全尺寸图片幻灯片

图 2 Osher-Shu问题的密度分布曲线

Figure 2. Density distribution curves of Osher-Shu problem

下载: 全尺寸图片幻灯片

图 3 二维Riemann问题的密度等值线

Figure 3. Density contours of 2D Riemann problem

下载: 全尺寸图片幻灯片

图 4 双马赫杆区域的密度等值线

Figure 4. Density contours at double Mach stems region

下载: 全尺寸图片幻灯片

表 1 Lax问题的计算耗时

Table 1. Computational time for Lax problem

格式	WENO-Z(N=100)	HCW-R(N=100)	HCW-E(N=100)	HCW-E(N=120)
计算耗时/s	2.03	3.20	1.16	1.61

下载: 导出CSV

表 2 Osher-Shu问题的计算耗时

Table 2. Computational time for Osher-Shu problem

格式	WENO-Z(N=200)	HCW-R(N=200)	HCW-E(N=200)	HCW-E(N=250)
计算耗时/s	7.81	17.57	6.36	9.83

下载: 导出CSV

表 3 二维Riemann问题的计算耗时

Table 3. Computational time for 2D Riemann problem

格式	WENO-Z(N=400×400)	HCW-R(N=400×400)	HCW-E(N=400×400)	HCW-E(N=600×600)
计算耗时/s	5188	23439	4473	17347

下载: 导出CSV

表 4 双马赫反射问题的计算耗时

Table 4. Computational time for double Mach reflection problem

格式	WENO-Z(N=960×240)	HCW-R(N=960×240)	HCW-E(N=960×240)	HCW-E(N=1 600×400)
计算耗时/s	7371	31460	6804	28118

下载: 导出CSV

参考文献(18)

[1]	LIU X D, OSHER S, CHAN T.Weighted essentially non-oscillatory schemes[J].Journal of Computational Physics, 1994, 115(1):200-212. doi: 10.1006/jcph.1994.1187
[2]	JIANG G S, SHU C W.Efficient implementation of weighted ENO schemes[J].Journal of Computational Physics, 1996, 126(1):202-228. doi: 10.1006-jcph.1996.0130/
[3]	HENRICK A K, ASLAM T D, POWERS J M.Mapped weighted essentially non-oscillatory schemes:Achieving optimal order near critical points[J].Journal of Computational Physics, 2005, 207(2):542-567. doi: 10.1016/j.jcp.2005.01.023
[4]	BORGES R, CARMONA M, COSTA B, et al.An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J].Journal of Computational Physics, 2008, 227(6):3191-3211. doi: 10.1016/j.jcp.2007.11.038
[5]	ACKER F, BORGES R, COSTA B, et al.An improved WENO-Z scheme[J].Journal of Computational Physics, 2016, 313(1):726-753. http://d.old.wanfangdata.com.cn/NSTLQK/NSTL_QKJJ021479294/
[6]	骆信, 吴颂平.改进的五阶WENO-Z+格式[J].力学学报, 2019, 51(6):1927-1939. http://d.old.wanfangdata.com.cn/Periodical/lxxb201906029 LUO X, WU S P.An improved fifth-order WENO-Z+ scheme[J].Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(6):1927-1939(in Chinese). http://d.old.wanfangdata.com.cn/Periodical/lxxb201906029
[7]	LELE S K.Compact finite difference schemes with spectral-like resolution[J].Journal of Computational Physics, 1992, 103(1):16-42. http://d.old.wanfangdata.com.cn/OAPaper/oai_arXiv.org_1301.2885
[8]	PIROZZOLI S.Conservative hybrid compact-WENO schemes for shock-turbulence interaction[J].Journal of Computational Physics, 2002, 178(1):81-117. https://www.sciencedirect.com/science/article/pii/S002199910297021X
[9]	REN Y X, LIU M E, ZHANG H X.A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws[J].Journal of Computational Physics, 2003, 192(2):365-386. https://www.sciencedirect.com/science/article/pii/S002199910300398X
[10]	王来, 吴颂平.无自由参数型混合格式[J].北京航空航天大学学报, 2015, 41(2):318-322. doi: 10.13700/j.bh.1001-5965.2014.0134 WANG L, WU S P.Hybrid finite difference schemes without free parameters[J].Journal of Beijing University of Aeronautics and Astronautics, 2015, 41(2):318-322(in Chinese). doi: 10.13700/j.bh.1001-5965.2014.0134
[11]	LI Y S, YAN C, YU J, et al.A new high-accuracy scheme for compressible turbulent flows[J].International Journal of Computational Fluid Dynamics, 2017, 31(9):362-378. doi: 10.1080/10618562.2017.1365844
[12]	PENG J, SHEN Y Q.A novel weighting switch function for uniformly high-order hybrid shock-capturing schemes[J].International Journal for Numerical Methods in Fluids, 2017, 83(9):681-703. doi: 10.1002/fld.4285
[13]	武从海, 赵宁, 田琳琳.一种改进的紧致WENO混合格式[J].空气动力学学报, 2013, 31(4):477-481. http://d.old.wanfangdata.com.cn/Periodical/kqdlxxb201304012 WU C H, ZHAO N, TIAN L L.An improved hybrid compact-WENO scheme[J].Acta Aerodynamica Sinica, 2013, 31(4):477-481(inChinese). http://d.old.wanfangdata.com.cn/Periodical/kqdlxxb201304012
[14]	FU D X, MA Y W.A high order accurate difference scheme for complex flow fields[J].Journal of Computational Physics, 1997, 134(1):1-15. doi: 10.1006-jcph.1996.5492/
[15]	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. doi: 10.1002/cpa.3160070112
[16]	SHU C W, OSHER S.Efficient implementation of essentially non-oscillatory shock-capturing schemes, Ⅱ[J].Journal of Computational Physics, 1989, 83(1):32-78. doi: 10.1007/978-3-642-60543-7_14
[17]	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. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=0c4383af1961db475fda9b4efdf679b4
[18]	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. doi: 10.1016-0021-9991(84)90142-6/