基于新的参考光滑性指示子的改进的三阶WENO格式

王亚辉

doi:10.21656/1000-0887.420194

基于新的参考光滑性指示子的改进的三阶WENO格式

doi: 10.21656/1000-0887.420194

王亚辉

郑州师范学院数学与统计学院，郑州 450044

基金项目: 国家自然科学基金(12071470)；河南省高等学校重点科研项目(22B110020)

详细信息

作者简介:
王亚辉(1990—)，男，博士(E-mail：wangyh14@lsec.cc.ac.cn)

中图分类号: O357.4
计量
- 文章访问数: 809
- HTML全文浏览量: 376
- PDF下载量: 107
- 被引次数: 0
出版历程
- 收稿日期: 2021-07-12
- 修回日期: 2021-08-22
- 刊出日期: 2022-07-15

An Improved 3rd-Order WENO Scheme Based on a New Reference Smoothness Indicator

WANG Yahui

School of Mathematics and Statistics, Zhengzhou Normal University, Zhengzhou 450044, P.R.China

摘要

摘要:
针对计算流体力学对高精度高分辨率的需求，基于降低经典的三阶加权本质无振荡(WENO)格式的数值耗散特性，该文提出了一种新的参考光滑性指示子。其构造方法与经典的WENO-Z格式不同，它是通过候选子模板上重构多项式的导数的线性组合与整个全局模板上重构多项式的导数的$ L^2$范数逼近获得的。采用该计算方法可以得到比WENO-Z格式更高阶的参考光滑性指示子，另外改变自由参数$ \varphi$的取值，可以获得不同的参考光滑性指示子。该文通过一系列数值算例证明了该参考光滑性指示子的有效性。
- 双曲守恒律 /
- WENO /
- 重构多项式 /
- L²范数 /
- 非线性权
Abstract:
In order to meet the requirement of high accuracy and high resolution in computational fluid dynamics (CFD), a new reference smoothness indicator was proposed to reduce the numerical dissipation of the classical 3rd-order weighted essentially non-oscillatory (WENO) scheme. The construction method is different from the classical WENO-Z scheme. It is obtained through the L²-norm approximation of the derivatives of the reconstruction polynomials of the whole global stencil, and the linear combination of the derivatives of the reconstruction polynomials on the candidate sub-stencils. With this calculation method, higher-order reference smoothness indicators can be obtained than the WENO-Z scheme. In addition, different reference smoothness indicators can be obtained by change of the value of free parameter $ \varphi$. A series of numerical examples prove the effectiveness of the reference smoothness indicator.
- hyperbolic conservation law /
- WENO /
- reconstruction polynomial /
- L²-norm /
- nonlinear weight

HTML全文

图 1 三阶WENO数值通量的模板

Figure 1. The stencils for the 3rd-order WENO numerical flux

下载: 全尺寸图片幻灯片

图 2 线性对流方程(20)在初值(22)下，不同格式的数值解与解析解的比较，$t=41$，$N=400$

Figure 2. Comparison of the analytical solution with the numerical solutions to linear advection eq. (20) with an initial value (22) at $t=41$, N = 400

下载: 全尺寸图片幻灯片

图 3 线性对流方程(20)在初值(22)下，WENO-Re3格式在不同参数$\varphi$下的数值解比较，$t=41$，$N=400$

Figure 3. Comparison of the numerical solutions of the WENO-Re3 scheme with different parameter $\varphi$ values of linear advection eq. (20) with an initial value (22) at $t=41$, $N=400$

下载: 全尺寸图片幻灯片

图 4 线性对流方程(20)在初值(23)下，不同格式的数值解与解析解的比较，$t=6$，$N=400$

Figure 4. Comparison of the analytical solution with the numerical solutions of linear advection eq. (20) with an initial value (23) at t=6, N=400

下载: 全尺寸图片幻灯片

图 6 Lax激波管问题^[20]的数值结果，$t=0.13$，$N=200$

Figure 6. Numerical results of the Lax problem^[20], $t=0.13$, $N=200$

下载: 全尺寸图片幻灯片

图 5 线性对流方程(20)在初值(23)下，不同格式的数值解与解析解的比较，$t=6$，$N=400$（局部放大图）

Figure 5. Comparison of the analytical solution with the numerical solutions to linear advection eq. (20) with initial value (23) at t=6, N=400(local encargement)

下载: 全尺寸图片幻灯片

图 7 冲击波相互作用的数值结果，$t=0.038$，$N=400$

Figure 7. Numerical results of interacting blast waves, $t=0.038$, $N=400$

下载: 全尺寸图片幻灯片

图 8 不同参数$\varphi$下，冲击波相互作用的数值结果，$t=0.038$，$N=400$

Figure 8. Numerical results of interacting blast waves with different parameter $\varphi$ values, $t=0.038$, $N=400$

下载: 全尺寸图片幻灯片

图 9 激波等熵波相互作用(Titarev-Toro)^[21]的密度分布，$t=5$，$N=4\;001$

Figure 9. Density profiles of the shock entropy interaction (Titarev-Toro)^[21], t=5, N=4 001

下载: 全尺寸图片幻灯片

图 10 不同格式关于二维Riemann问题^[22]的密度等值线，$\Delta {{x}}=\Delta {{y}}=1/800$，$t=0.8$

Figure 10. Density contours of the 2D Riemann problem^[22], $\Delta {{x}}=\Delta {{y}}=1/800$, $t=0.8$

下载: 全尺寸图片幻灯片

图 11 双Mach反射问题在$t=0.2$时的密度等值线，网格点为$1\;920\times480$

Figure 11. Density contours of the double Mach reflection problem for $t=0.2$ on the $1\;920\times480$ grid points

下载: 全尺寸图片幻灯片

表 1 线性对流方程(20)在初值(21a)下，不同格式在$t = 2$的L¹误差和收敛阶

Table 1. L¹ errors and convergence rates at $t = 2$ of different schemes for linear advection eq. (20) with initial data (21a)

$ N $	WENO-JS3	WENO-Z3	WENO-P3	WENO-Re3
$ N $	$ L^{1} $ error (order)	$ L^{1} $ error (order)	$ L^{1} $ error (order)	$ L^{1} $ error (order)
$ 10 $	0.30(–)	0.22(–)	0.17(–)	0.12(–)
$ 20 $	9.07E − 2(1.73)	7.31E − 2(1.59)	4.94E − 2(1.78)	2.78E − 2(2.11)
$ 40 $	3.83E − 2(1.24)	2.06E − 2(1.83)	1.19E − 2(2.05)	6.67E − 3(2.06)
$ 80 $	9.62E − 3(1.99)	4.85E − 3(2.09)	2.53E − 3(2.23)	1.48E − 3(2.17)
$ 160 $	2.33E − 3(2.05)	1.06E − 3(2.19)	5.29E − 4(2.26)	2.98E − 4(2.31)
$ 320 $	5.46E − 4(2.09)	2.18E − 4(2.28)	1.03E − 4(2.36)	5.95E − 5(2.32)
$ 640 $	1.23E − 4(2.15)	3.94E − 5(2.47)	2.01E − 5(2.36)	1.13E − 5(2.40)

下载: 导出CSV

表 2 线性对流方程(20)在初值(21a)下，不同格式在$ t = 2 $的L^∞误差和收敛阶

Table 2. L^∞ errors and convergence rates at $ t = 2 $ of different schemes for linear advection eq. (20) with initial data (21a)

$ N $	WENO-JS3	WENO-Z3	WENO-P3	WENO-Re3
$ N $	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)
$ 10 $	0.53(–)	0.43(–)	0.35(–)	0.27(–)
$ 20 $	0.21(1.33)	0.15(1.52)	0.11(1.67)	7.84E − 2(1.78)
$ 40 $	8.76E − 2(1.26)	5.94E − 2(1.34)	4.08E − 2(1.43)	2.75E − 2(1.51)
$ 80 $	3.51E − 2(1.32)	2.23E − 2(1.41)	1.45E − 2(1.49)	9.71E − 3(1.50)
$ 160 $	1.36E − 2(1.37)	8.16E − 3(1.45)	5.05E − 3(1.52)	3.38E − 3(1.52)
$ 320 $	5.19E − 3(1.39)	2.86E − 3(1.51)	1.73E − 3(1.55)	1.15E − 3(1.56)
$ 640 $	1.91E − 3(1.44)	8.94E − 4(1.68)	5.81E − 4(1.57)	3.87E − 4(1.57)

下载: 导出CSV

表 3 线性对流方程(20)在初值(21b)下，不同格式在$ t = 2$的L¹误差和收敛阶

Table 3. L¹ errors and convergence rates at $ t = 2 $ of different schemes for linear advection eq. (20) with initial data (21b)

$ N $	WENO-JS3	WENO-Z3	WENO-P3	WENO-Re3
$ N $	$ L^{1} $ error (order)	$ L^{1} $ error (order)	$ L^{1} $ error (order)	$ L^{1} $ error (order)
$ 10 $	0.29(–)	0.23(–)	0.18(–)	0.15(–)
$ 20 $	1.14E − 1(1.35)	7.74E − 2(1.57)	5.45E − 2(1.72)	3.49E − 2(2.10)
$ 40 $	4.21E − 2(1.44)	2.40E − 2(1.69)	1.35E − 2(2.01)	7.90E − 3(2.14)
$ 80 $	1.13E − 2(1.90)	5.74E − 3(2.06)	3.10E − 3(2.12)	1.84E − 3(2.10)
$ 160 $	2.76E − 3(2.03)	1.28E − 3(2.16)	6.45E − 4(2.26)	3.84E − 4(2.26)
$ 320 $	6.53E − 4(2.08)	2.68E − 4(2.26)	1.28E − 4(2.33)	7.37E − 5(2.38)
$ 640 $	1.47E − 4(2.15)	4.89E − 5(2.45)	2.44E − 5(2.39)	1.40E − 5(2.40)

下载: 导出CSV

表 4 线性对流方程(20)在初值(21b)下，不同格式在$ t = 2 $的L^∞误差和收敛阶

Table 4. L^∞ errors and convergence rates at $ t = 2 $ of different schemes for linear advection eq. (20) with initial data (21b)

$ N $	WENO-JS3	WENO-Z3	WENO-P3	WENO-Re3
$ N $	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)
$ 10 $	0.55(–)	0.45(–)	0.36(–)	0.30(–)
$ 20 $	2.47E − 1(1.15)	1.76E − 1(1.35)	1.34E − 1(1.43)	9.74E − 2(1.62)
$ 40 $	1.00E − 1(1.30)	7.03E − 2(1.32)	4.97E − 2(1.43)	3.43E − 2(1.51)
$ 80 $	4.19E − 2(1.25)	2.69E − 2(1.39)	1.78E − 2(1.48)	1.21E − 2(1.50)
$ 160 $	1.64E − 2(1.35)	9.92E − 3(1.44)	6.22E − 3(1.52)	4.24E − 3(1.51)
$ 320 $	6.27E − 3(1.39)	3.50E − 3(1.50)	2.13E − 3(1.55)	1.44E − 3(1.56)
$ 640 $	2.30E − 3(1.45)	1.10E − 3(1.67)	7.00E − 4(1.61)	4.67E − 4(1.62)

下载: 导出CSV

参考文献(23)

[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]	HARTEN A, OSHER S. Uniformly high-order accurate non-oscillatory schemes Ⅰ[J]. SIAM Journal on Numerical Analysis, 1987, 24(2): 279-309. doi: 10.1137/0724022
[3]	HARTEN A, ENGQUIST B, OSHER S, et al. Uniformly high-order accurate essentially non-oscillatory schemes Ⅲ[J]. Journal of Computational Physics, 1987, 71: 231-303. doi: 10.1016/0021-9991(87)90031-3
[4]	SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes[J]. Journal of Computational Physics, 1988, 77(2): 439-471. doi: 10.1016/0021-9991(88)90177-5
[5]	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: 10.1016/0021-9991(89)90222-2
[6]	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
[7]	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
[8]	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: 3191-3211. doi: 10.1016/j.jcp.2007.11.038
[9]	GEROLYMOS R A, SÉNÉCHAL S, VALLET I. Very-high-order WENO schemes[J]. Journal of Computational Physics, 2009, 228(23): 8481-8524. doi: 10.1016/j.jcp.2009.07.039
[10]	WANG Y H, DU Y L, ZHAO K L, et al. Modified stencil approximations for fifth-order weighted essentially non-oscillatory schemes[J]. Journal of Scientific Computing, 2019, 81(6): 898-922.
[11]	FU L, HU X Y, ADAMS N A. A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 2018, 374: 724-751. doi: 10.1016/j.jcp.2018.07.043
[12]	WU X S, ZHAO Y X. A high-resolution hybrid scheme for hyperbolic conservation laws[J]. International Journal for Numerical Methods in Fluids, 2015, 78(3): 162-187. doi: 10.1002/fld.4014
[13]	WU X S, LIANG J H, ZHAO Y X. A new smoothness indicator for third-order WENO scheme[J]. International Journal for Numerical Methods in Fluids, 2016, 81(7): 451-459. doi: 10.1002/fld.4194
[14]	XU W Z, WU W G. An improved third-order WENO-Z scheme[J]. Journal of Scientific Computing, 2018, 75: 1808-1841. doi: 10.1007/s10915-017-0587-4
[15]	WANG Y H, DU Y L, ZHAO K L, et al. A low-dissipation third-order weighted essentially nonoscillatory scheme with a new reference smoothness indicator[J]. International Journal for Numerical Methods in Fluid, 2020, 92(9): 1212-1234.
[16]	王亚辉. 求解双曲守恒律方程的三阶修正模板WENO格式[J]. 应用数学和力学, 2022, 43(2): 224-236. (WANG Yahui. A 3rd-order modified stencil WENO scheme for solution of hyperbolic conservation law equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 224-236.(in Chinese) WANG Yahui. A 3rd-order modified stencil WENO scheme for solution of hyperbolic conservation law equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 224-236. (in Chinese))
[17]	徐维铮, 孔祥韶, 吴卫国. 基于映射函数的三阶 WENO 改进格式及其应用[J]. 应用数学和力学, 2017, 38(10): 1120-1135. (XU Weizheng, KONG Xiangshao, WU Weiguo. An improved 3rd-order WENO scheme based on mapping functions and its application[J]. Applied Mathematics and Mechanics, 2017, 38(10): 1120-1135.(in Chinese) XU Weizheng, KONG Xiangshao, WU Weiguo. An improved 3rd-order WENO scheme based on mapping functions and its application[J]. Applied Mathematics and Mechanics, 2017, 38(10): 1120-1135. (in Chinese)
[18]	徐维铮, 吴卫国. 三阶WENO-Z格式精度分析及其改进格式[J]. 应用数学和力学, 2018, 39(8): 946-960. (XU Weizheng, WU Weiguo. Precision analysis of the 3rd-order WENO-Z scheme and its improved scheme[J]. Applied Mathematics and Mechanics, 2018, 39(8): 946-960.(in Chinese) XU Weizheng, WU Weiguo. Precision analysis of the 3rd-order WENO-Z scheme and its improved scheme[J]. Applied Mathematics and Mechanics, 2018, 39(8): 946-960. (in Chinese))
[19]	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
[20]	SOD G A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws[J]. Journal of Computational Physics, 1978, 27(1): 1-31. doi: 10.1016/0021-9991(78)90023-2
[21]	TITAREV V A, TORO E F. Finite-volume WENO schemes for three-dimensional conservation laws[J]. Journal of Computational Physics, 2004, 201(1): 238-260. doi: 10.1016/j.jcp.2004.05.015
[22]	SCHULZ-RINNE C W, COLLINS J P, GLAZ H M. Numerical solution of the Riemann problem for two-dimensional gas dynamics[J]. SIAM Journal on Scientific Computing, 1993, 14(6): 1394-1414. doi: 10.1137/0914082
[23]	WOODWAED P, COLELLA P. The numerical simulation of two-dimensional fluid flow with strong shocks[J]. Journal of Computational Physics, 1984, 54(1): 447-465.