求解双曲守恒律方程的三阶修正模板WENO格式

王亚辉

doi:10.21656/1000-0887.420091

求解双曲守恒律方程的三阶修正模板WENO格式

doi: 10.21656/1000-0887.420091

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: O357.41
计量
- 文章访问数: 1154
- HTML全文浏览量: 533
- PDF下载量: 99
- 被引次数: 0
出版历程
- 收稿日期: 2021-04-10
- 录用日期: 2021-04-10
- 修回日期: 2021-09-09
- 网络出版日期: 2021-12-27
- 刊出日期: 2022-02-01

A 3rd-Order Modified Stencil WENO Scheme for Solution of Hyperbolic Conservation Law Equations

WANG Yahui

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

摘要

摘要:
为了降低经典的三阶加权本质无振荡(WENO)格式的数值耗散，提出了一种新的三阶WENO格式的修正模板近似方法。改进了经典WENO-JS格式中各候选模板上数值通量的一阶多项式逼近，通过加入二次项使模板逼近达到三阶精度。计算了相应的候选通量，并且通过引入可调函数φ(x)，使得新的格式具有ENO性质。最后给出了一系列数值算例，证明了该方法的有效性。
- 双曲守恒律 /
- WENO /
- 修正模板 /
- 非线性权
Abstract:
In order to reduce the numerical dissipation of the classical 3rd-order weighted essentially non-oscillatory (WENO) scheme, a new modified stencil approximation of the 3rd-order WENO scheme was proposed. The 1st-order polynomial approximation of numerical flux on each candidate stencil in the classical WENO-JS3 scheme was improved, and the quadratic term was added to make the stencil approximation reach the 3rd-order accuracy. The corresponding candidate fluxes were calculated. Moreover, the new scheme has essentially non-oscillatory properties through introduction of tunable function φ(x). A series of numerical examples show the effectiveness of the new method.
- hyperbolic conservation law /
- WENO /
- modified stencil /
- nonlinear weight

HTML全文

图 1 三阶WENO数值通量的模板

Figure 1. 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(31)在初值(33)下数值解与解析解的比较，$t=40$

Figure 2. Comparison of the analytical solution with the numerical solutions of linear advection eq. (31) with initial value (33), at $t=40$

下载: 全尺寸图片幻灯片

图 3 线性对流方程(31)在初值(34)下不同格式的数值解与解析解的比较，$t=41$，$N=400$

Figure 3. Comparison of the analytical solution with the numerical solutions of linear advection eq. (31) with initial value (34), at t=41, N=400

下载: 全尺寸图片幻灯片

图 4 线性对流方程(31)在初值(35)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. (31) with initial value (35), at $t=6, \;N=400$

下载: 全尺寸图片幻灯片

图 5 Sod激波管问题^[16]的数值结果，$t=0.2$，$N=200$，$C_{\rm{CFL}}=0.6$

Figure 5. Numerical results of the Sod problem^[16], at $t=0.2$，$N=200$，$C_{\rm{CFL}}=0.6$

下载: 全尺寸图片幻灯片

图 6 冲击波的相互作用的数值结果，$t=0.038$，$N=800$，$C_{\rm{CFL}}=0.6$

Figure 6. Numerical results of interacting blast waves, at $t=0.038$，$N=800$，C_CFL = 0.6

下载: 全尺寸图片幻灯片

图 7 激波等熵波相互作用(Shu-Osher)^[5]的密度分布，$t=1.8$，$N=401$，$C_{\rm{CFL}}=0.6$

Figure 7. Density profiles of the shock entropy interacting of Shu-Osher^[5] at $t=1.8$ with 401 points and $C_{\rm{CFL}}=0.6$

下载: 全尺寸图片幻灯片

图 8 不同格式关于Rayleigh-Taylor不稳定性问题的密度等值线，$\Delta x=\Delta y=1/240$，$C_{\rm{CFL}}=0.6$，$t=1.95$

Figure 8. Density contours of the Rayleigh-Taylor instability at $t=1.95$ with $\Delta x=\Delta y=1/240$，$C_{\rm{CFL}}=0.6$

下载: 全尺寸图片幻灯片

图 9 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 9. Density contours of the double Mach reflection problem at $t=0.2$ with $1\;920\times480$ grid points

下载: 全尺寸图片幻灯片

表 1 线性对流方程(31)在初值(32a)下，不同格式在$ t=2.0 $的L¹误差和收敛阶

Table 1. $ L^1 $ errors and convergence rates at $ t=2.0 $ of different schemes for linear advection eq. (31) with initial data eq. (32a)

$ N $	WENO-JS3	WENO-Z3	WENO-MS-JS3	WENO-MS-Z3
$ N $	$ L^{1} $ error (order)	$ L^{1} $ error (order)	$ L^{1} $ error (order)	$ L^{1} $ error (order)
$ 10 $	2.99E−1(−)	2.21E−1(−)	1.82E−1(−)	1.59E−1(−)
$ 20 $	9.07E−2(1.72)	7.31E−2(1.60)	5.84E−2(1.64)	4.80E−2(1.73)
$ 40 $	3.83E−2(1.24)	2.06E−2(1.83)	1.42E−2(2.04)	1.16E−2(2.05)
$ 80 $	9.62E−3(1.99)	4.85E−3(2.09)	3.14E−3(2.18)	2.45E−3(2.24)
$ 160 $	2.33E−3(2.05)	1.06E−3(2.19)	6.53E−4(2.27)	5.11E−4(2.26)
$ 320 $	5.46E−4(2.09)	2.18E−4(2.28)	1.17E−4(2.48)	8.12E−5(2.65)
$ 640 $	1.23E−4(2.15)	3.94E−5(2.47)	1.44E−5(3.02)	8.93E−6(3.18)
$1\;280$	2.43E−5(2.34)	5.55E−6(2.83)	1.78E−6(3.02)	1.05E−6(3.09)

下载: 导出CSV

表 2 线性对流方程(31)在初值(32a)下，不同格式在$ t=2.0 $的$ L^{\infty} $误差和收敛阶

Table 2. $ L^{\infty} $ errors and convergence rates at $ t=2.0 $ of different schemes for linear advection eq. (31) with initial data eq. (32a)

$ N $	WENO-JS3	WENO-Z3	WENO-MS-JS3	WENO-MS-Z3
$ N $	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)
$ 10 $	5.30E−1(−)	4.31E−1(−)	3.80E−1(−)	3.43E−1(−)
$ 20 $	2.10E−1(1.34)	1.52E−1(1.50)	1.26E−1(1.59)	1.11E−1(1.63)
$ 40 $	8.76E−2(1.26)	5.94E−2(1.36)	4.67E−2(1.43)	4.02E−2(1.47)
$ 80 $	3.51E−2(1.32)	2.23E−2(1.41)	1.67E−2(1.48)	1.42E−2(1.50)
$ 160 $	1.36E−2(1.37)	8.16E−3(1.45)	5.85E−3(1.51)	4.94E−3(1.52)
$ 320 $	5.19E−3(1.39)	2.86E−3(1.51)	1.87E−3(1.65)	1.43E−3(1.79)
$ 640 $	1.91E−3(1.44)	8.94E−4(1.68)	4.37E−4(2.10)	2.73E−4(2.39)
$1\;280$	6.38E−4(1.58)	2.25E−4(1.99)	5.47E−5(3.00)	3.35E−5(3.03)

下载: 导出CSV

表 3 线性对流方程(31)在初值(32b)下，不同格式在$ t=2.0 $的L¹误差和收敛阶

Table 3. $ L^1 $ errors and convergence rates at $ t=2.0 $ of different schemes for linear advection eq. (31) with initial data eq. (32b)

$ N $	WENO-JS3	WENO-Z3	WENO-MS-JS3	WENO-MS-Z3
$ N $	$ L^{1} $ error (order)	$ L^{1} $ error (order)	$ L^{1} $ error (order)	$ L^{1} $ error (order)
$ 10 $	2.94E−1(−)	2.31E−1 (−)	1.99E−1 (−)	1.79E−1 (−)
$ 20 $	1.14E−1(1.37)	7.74E−2(1.58)	6.31E−2(1.66)	5.33E−2(1.75)
$ 40 $	4.21E−2(1.44)	2.40E−2(1.69)	1.63E−2(1.95)	1.31E−2(2.01)
$ 80 $	1.13E−2(1.90)	5.74E−3(2.06)	3.81E−3(2.10)	3.02E−3(2.13)
$ 160 $	2.76E−3(2.03)	1.28E−3(2.16)	7.95E−4(2.26)	6.17E−4(2.29)
$ 320 $	6.53E−4(2.08)	2.68E−4(2.26)	1.48E−4(2.43)	1.04E−4(2.57)
$ 640 $	1.47E−4(2.15)	4.89E−5(2.45)	1.91E−5(2.95)	1.10E−5(3.24)
$1\;280$	3.02E−5(2.28)	7.12E−6(2.78)	2.36E−6(3.02)	1.37E−6(3.01)

下载: 导出CSV

表 4 线性对流方程(31)在初值(32b)下，不同格式在$ t=2.0 $的L^∞误差和收敛阶

Table 4. $ L^{\infty} $ errors and convergence rates at $ t=2.0 $ of different schemes for linear advection eq. (31) with initial data eq. (32b)

$ N $	WENO-JS3	WENO-Z3	WENO-MS-JS3	WENO-MS-Z3
$ N $	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)	$ L^{\infty} $ error (order)
$ 10 $	5.46E−1(−)	4.49E−1 (−)	3.96E−1 (−)	3.61E−1 (−)
$ 20 $	2.47E−1(1.44)	1.76E−1(1.35)	1.48E−1(1.42)	1.31E−1(1.46)
$ 40 $	1.04E−1(1.25)	7.03E−2(1.32)	5.58E−2(1.41)	4.86E−2(1.43)
$ 80 $	4.19E−2(1.31)	2.69E−2(1.39)	2.04E−2(1.45)	1.74E−2(1.48)
$ 160 $	1.64E−2(1.35)	9.92E−3(1.44)	7.20E−3(1.50)	6.03E−3(1.53)
$ 320 $	6.27E−3(1.39)	3.50E−3(1.50)	2.34E−3(1.62)	1.82E−3(1.73)
$ 640 $	2.30E−3(1.45)	1.10E−3(1.67)	5.62E−4(2.05)	3.62E−4(2.33)
$1\;280$	7.82E−4(1.56)	2.88E−4(1.93)	7.87E−5(2.83)	4.76E−5(2.93)

下载: 导出CSV

参考文献(20)

[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 of 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(2): 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: 542-567. doi: 10.1016/j.jcp.2005.01.023
[8]	BORGES R, CARMONA M, COSTA B, et al. An improved WENO scheme for hyperbolic conservation laws[J]. Journal of Computational Physics, 2008, 227: 3191-3211. doi: 10.1016/j.jcp.2007.11.038
[9]	GEROLYMOS G A, S’EN’ECHAL D, 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]	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
[11]	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
[12]	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
[13]	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: 1212-1234. doi: 10.1002/fld.4824
[14]	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)
[15]	徐维铮, 吴卫国. 三阶WENO-Z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)
[16]	SHU C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws: NASA/CR-97-206253[R]. ICASE Report, 1997.
[17]	SOD G A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws[J]. Journal of Computational Physics, 1978, 27: 1-31. doi: 10.1016/0021-9991(78)90023-2
[18]	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
[19]	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
[20]	YOUNG Y N, TUFO H, DUBEY A, et al. On the miscible Rayleigh-Taylor instability: two and three dimensions[J]. Journal of Fluid Mechanics, 2001, 447: 377-408. doi: 10.1017/S0022112001005870