基于映射的模板光滑探测子的三阶WENO格式

王亚辉; 郭城; 杜玉龙

doi:10.21656/1000-0887.450150

基于映射的模板光滑探测子的三阶WENO格式

doi: 10.21656/1000-0887.450150

王亚辉^1, ,,
郭城^1,,
杜玉龙^2,

1.
郑州师范学院数学与统计学院，郑州 450044
2.
江苏海洋大学理学院，江苏连云港 222005

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

河南省自然科学基金 252300420394

详细信息

作者简介:
郭城(1980—)，男，硕士(E-mail: gc_scv@163.com)

杜玉龙(1988—)，男，博士(E-mail: kunyu0918@163.com)

通讯作者:
王亚辉(1991—)，男，博士(通讯作者. E-mail: hlg_cfd2014@163.com)

中图分类号: O357.41
计量
- 文章访问数: 86
- HTML全文浏览量: 34
- PDF下载量: 21
- 被引次数: 0
出版历程
- 收稿日期: 2024-05-21
- 修回日期: 2024-07-06
- 刊出日期: 2025-03-01

A 3rd-Order WENO Scheme for Stencil Smoothness Indicators Based on Mapping

WANG Yahui^{1
, ,},
GUO Cheng^1
,,
DU Yulong^2
,

1.
School of Mathematics and Statistics, Zhengzhou Normal University, Zhengzhou 450044, P.R.China
2.
School of Science, Jiangsu Ocean University, Lianyungang, Jiangsu 222005, P.R.China

摘要

摘要: 加权本质无振荡(weighted essentially non-oscillatory, WENO)格式能否具有低耗散特性及高分辨率特性, 关键在于光滑探测子的构造.该文针对三阶WENO格式的光滑探测子进行修正, 通过最光滑的探测子, 构造出了一个关于子模板光滑探测子的映射函数.在该函数作用下, 减小了欠光滑模板的光滑探测子, 进而增大了欠光滑模板的非线性权重.这明显地降低了格式的数值耗散, 提高了格式的分辨率.一系列数值测试表明，基于映射的模板光滑探测子的三阶WENO格式比传统的三阶WENO-JS3和WENO-Z3格式具有更高的分辨率.
- 双曲守恒律 /
- WENO /
- 映射 /
- 光滑探测子 /
- 非线性权
Abstract: The key to whether the WENO scheme can achieve optimal convergence accuracy and maintain essential no-oscillation characteristics near discontinuities lies in the construction of smoothness indicators. The smoothness indicator of the 3rd-order WENO scheme was modified through the construction of a mapping function to correct the smoothness indicators on each candidate stencil with the smoothest indicator. Under the influence of this mapping function, the smoothness indicator of the under-smooth stencil was reduced, thereby the nonlinear weight of the under-smooth stencil was increased. Then the numerical dissipation of the scheme was significantly lowered and its resolution was improved. A series of numerical examples demonstrate that, the new 3rd-order WENO scheme for smoothness indicators based on mapping has higher resolution than the classical WENO-JS3 and WENO-Z3 schemes.
- hyperbolic conservation law /
- WENO /
- mapping /
- smoothness indicator /
- nonlinear weight

HTML全文

图 1 三阶WENO数值通量模板

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

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

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图 2 冲击波相互作用的数值结果(t=0.038, N=400)

Figure 2. Numerical results of interacting blast waves for t=0.038, N=400

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图 3 冲击波相互作用的数值结果(t=0.038, N=800)

Figure 3. Numerical results of interacting blast waves for t=0.038, N=800

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图 4 线性对流方程(25)在初值(27)下，不同格式的数值解与解析解的比较(t=41, N=400)

Figure 4. Comparison of the analytical solution with the numerical solutions of linear advection eq. (25) with initial data (27) for t=41, N=400

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图 5 线性对流方程(25)在初值(28)下，不同格式的数值解与解析解的比较(t=400, N=400)

Figure 5. Comparison of the analytical solution with the numerical solutions of linear advection eq. (25) with initial data (28) for t=400, N=400

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图 6 线性对流方程(25)在初值(29)下，不同格式的数值解与解析解的比较(t=8, N=400)

Figure 6. Comparison of the analytical solution with the numerical solutions of linear advection eq. (25) with initial data (29) for t=8, N=400

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图 7 Sod激波管问题^[24]的数值结果(t=2.0, N=200)

Figure 7. Numerical results of the Sod problem^[24] for t=2.0, N=200

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图 8 Lax激波管问题^[24]的数值结果(t=1.3, N=200)

Figure 8. Numerical results of the Lax problem^[24] for t=1.3, N=200

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图 9 123激波管问题^[24]的数值结果(t=1.0, N=200)

Figure 9. Numerical results of the 123 problem^[24] for t=1.0, N=200

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图 10 冲击波的相互作用的数值结果(t=0.038, N=400)

Figure 10. Numerical results of interacting blast waves for t=0.038, N=400

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图 11 Sedov激波管问题^[24]的数值结果(t=0.004, N=480)

Figure 11. Numerical results of the Sedov problem^[24] for t=0.004, N=480

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图 12 激波等熵波相互作用(Shu-Osher)^[24-25]的密度分布(t=1.8, N=401)

Figure 12. Density profiles of the shock entropy interacting of Shu-Osher^[24-25] for t=1.8, N=401

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图 13 LeBlanc激波管问题的密度分布(t=6.0, N=3 201)

Figure 13. Density profiles of the LeBlanc shock tube problem for t=6.0, N=3 201

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图 14 二维Riemann问题的密度分布(t=0.8, N=801)

Figure 14. Density contours of 2D problems on 801×801 grid points for t=0.8, N=801

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图 15 Mach 3前台阶流问题的密度分布(t=4.0, Δx=Δy=200)

Figure 15. Density contours for the Mach 3 wind tunnel flow with a forward step for t=4.0, Δx=Δy=200

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表 1 线性对流方程(25)在初值(26a)下, 不同格式在t=2.0时的L¹误差和收敛阶

Table 1. L¹ errors and convergence rates with t=2.0 of different schemes for linear advection eq. (25) with initial data (26a)

N	WENO-JS3	WENO-Z3	WENO-JS3-M	WENO-Z3-M
N	L¹ error (order)	L¹ error (order)	L¹ error (order)	L¹ error (order)
25	7.60E-2(-)	5.31E-2(-)	6.88E-2(-)	3.93E-2(-)
50	2.45E-2(1.63)	1.31E-2(2.02)	1.73E-2(1.99)	9.25E-3(2.09)
100	5.82E-3(2.07)	2.90E-3(2.18)	3.26E-3(2.41)	1.97E-3(2.23)
200	1.05E-3(2.47)	5.56E-4(2.38)	3.65E-4(3.16)	3.90E-4(2.34)
400	2.24E-4(2.23)	7.82E-5(2.83)	4.67E-5(2.97)	6.32E-5(2.63)
800	3.20E-5(2.81)	1.45E-5(2.43)	5.09E-6(3.20)	6.76E-6(3.22)

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表 2 线性对流方程(25)在初值(26a)下, 不同格式在t=2.0时的L^∞误差和收敛阶

Table 2. L^∞ errors and convergence rates with t=2.0 of different schemes for linear advection eq. (25) with initial data (26a)

N	WENO-JS3	WENO-Z3	WENO-JS3-M	WENO-Z3-M
N	L^∞ error (order)	L^∞ error (order)	L^∞ error (order)	L^∞ error (order)
25	1.62E-1(-)	1.17E-1(-)	1.39E-1(-)	9.77E-2(-)
50	6.65E-2(1.28)	4.49E-2(1.38)	5.34E-2(1.38)	3.58E-2(1.45)
100	2.45E-2(1.44)	1.59E-2(1.50)	1.66E-2(1.69)	1.22E-2(1.55)
200	7.18E-3(1.77)	5.16E-3(1.62)	3.49E-3(2.25)	4.16E-3(1.55)
400	2.54E-3(1.50)	1.31E-3(1.98)	4.71E-4(2.89)	1.22E-3(1.77)
800	4.10E-4(2.63)	4.21E-4(1.64)	1.51E-4(1.64)	2.45E-4(2.32)

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表 3 线性对流方程(25)在初值(26b)下, 不同格式在t=2.0时的L¹误差和收敛阶

Table 3. L¹ errors and convergence rates with t=2.0 of different schemes for linear advection eq. (25) with initial data (26b)

N	WENO-JS3	WENO-Z3	WENO-JS3-M	WENO-Z3-M
N	L¹ error (order)	L¹ error (order)	L¹ error (order)	L¹ error (order)
25	8.18E-2(-)	5.47E-2(-)	2.91E-2(-)	1.82E-2(-)
50	2.88E-2(1.51)	1.50E-2(1.87)	8.37E-3(1.80)	4.65E-3(1.97)
100	7.19E-3(2.00)	3.27E-3(2.20)	1.99E-3(2.07)	1.14E-3(2.03)
200	1.63E-3(2.14)	5.32E-4(2.62)	2.47E-4(3.01)	2.16E-4(2.40)
400	2.88E-4(2.50)	1.02E-4(2.38)	4.40E-5(2.49)	2.80E-5(2.95)
800	3.08E-5(3.23)	9.35E-6(3.45)	5.47E-6(3.01)	3.87E-6(2.86)

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表 4 线性对流方程(25)在初值(26b)下, 不同格式在t=2.0时的L^∞误差和收敛阶

Table 4. L^∞ errors and convergence rates with t=2.0 of different schemes for linear advection eq. (25) with initial data (26b)

N	WENO-JS3	WENO-Z3	WENO-JS3-M	WENO-Z3-M
N	L^∞ error (order)	L^∞ error (order)	L^∞ error (order)	L^∞ error (order)
25	1.88E-1(-)	1.31E-1(-)	1.70E-1(-)	1.09E-1(-)
50	7.83E-2(1.26)	5.15E-2(1.35)	6.19E-2(1.46)	4.08E-2(1.42)
100	3.09E-2(1.34)	1.81E-2(1.51)	2.06E-2(1.59)	1.47E-2(1.47)
200	1.12E-2(1.46)	5.11E-3(1.82)	4.64E-3(2.15)	4.81E-3(1.61)
400	3.27E-3(1.78)	1.69E-3(1.60)	1.46E-3(1.67)	1.16E-3(2.05)
800	5.87E-4(2.48)	2.79E-4(2.60)	2.08E-4(2.81)	1.51E-4(2.94)

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表 5 不同WENO格式关于若干一维Riemann问题的时间成本

Table 5. The time costs of a number of 1D Riemann problems with different WENO schemes

problem	WENO-JS3	WENO-Z3	WENO-R3	WENO-JS3-M	WENO-Z3-M
Sod	0.146 5(1.00)	0.146 9(1.00)	0.175 5(1.20)	0.223 0(1.52)	0.224 9(1.53)
Lax	0.195 3(1.00)	0.193 0(0.99)	0.230 9(1.18)	0.303 9(1.55)	0.305 1(1.56)
Shu-Osher	0.261 1(1.00)	0.266 6(1.02)	0.316 0(1.21)	0.413 1(1.58)	0.414 7(1.59)
blast	0.577 6(1.00)	0.599 3(1.04)	0.706 5(1.22)	0.900 3(1.56)	0.912 9(1.58)
LeBlanc	9.414 0(1.00)	9.648 9(1.02)	11.559 4(1.23)	14.681 1(1.56)	14.889 6(1.58)

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