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基于新的参考光滑性指示子的改进的三阶WENO格式

王亚辉

王亚辉. 基于新的参考光滑性指示子的改进的三阶WENO格式 [J]. 应用数学和力学,2022,43(7):1-14 doi: 10.21656/1000-0887.420194
引用本文: 王亚辉. 基于新的参考光滑性指示子的改进的三阶WENO格式 [J]. 应用数学和力学,2022,43(7):1-14 doi: 10.21656/1000-0887.420194
Yahui WANG. An Improved Third Order WENO Scheme Based on a New Reference Smoothness Indicator[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420194
Citation: Yahui WANG. An Improved Third Order WENO Scheme Based on a New Reference Smoothness Indicator[J]. Applied Mathematics and Mechanics. 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
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

An Improved Third Order WENO Scheme Based on a New Reference Smoothness Indicator

  • 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。
  • 图  1  三阶WENO数值通量的模板

    Figure  1.  Stencils for the third-order WENO numerical flux

    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 of the linear advection eq. (20) with initial value(22) at $t=41$, N = 400

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

    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 the linear advection eq. (20) with initial value(23) at $t=6$ with 400 grid points

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

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

    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 of the linear advection eq. (20) with initial value(23) at $t=6$ with 400 grid points(local encarged drawing)

    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 at $t=0.038$, $N=400$

    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 the different parameter $\varphi$ at $t=0.038$, $N=400$

    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 interacting of Titarev-Toro[21]at $t=5$ with 4001 points

    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$:(a)WENO-JS3;(b)WENO-Z3;(c)WENO-P3;(d)WENO-Re3

    Figure  10.  Density contours of 2D Riemann Problem[22] on $801\times801$ grid points at $t=0.8$: (a) WENO-JS3; (b) WENO-Z3; (c) WENO-P3; (d) WENO-Re3

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

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

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

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

    $ N $WENO-JS3WENO-Z3WENO-P3WENO-Re3
    $ L^{1} $ error (order)$ L^{1} $ error (order)$ L^{1} $ error (order)$ L^{1} $ error (order)
    $ 10 $0.30E − 0(–)0.22E − 0(–)0.17E − 0(–)0.12E − 0(–)
    $ 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 $$ L^{\infty} $误差和收敛阶

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

    $ N $WENO-JS3WENO-Z3WENO-P3WENO-Re3
    $ L^{\infty} $ error (order)$ L^{\infty} $ error (order)$ L^{\infty} $ error (order)$ L^{\infty} $ error (order)
    $ 10 $0.53E − 0(–)0.43E − 0(–)0.35E − 0(–)0.27E − 0(–)
    $ 20 $0.21E − 0(1.33)0.15E − 0(1.52)0.11E − 0(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$L1误差和收敛阶

    Table  3.   $ L^1 $ errors and convergence rates at $ t = 2 $ of different schemes for the linear advection eq. (20) with the initial data (21b)

    $ N $WENO-JS3WENO-Z3WENO-P3WENO-Re3
    $ L^{1} $ error (order)$ L^{1} $ error (order)$ L^{1} $ error (order)$ L^{1} $ error (order)
    $ 10 $0.29E − 0(–)0.23E − 0 (–)0.18E − 0 (–)0.15E − 0 (–)
    $ 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 $的误差和收敛阶

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

    $ N $WENO-JS3WENO-Z3WENO-P3WENO-Re3
    $ L^{\infty} $ error (order)$ L^{\infty} $ error (order)$ L^{\infty} $ error (order)$ L^{\infty} $ error (order)
    $ 10 $0.55E − 0(–)0.45E − 0 (–)0.36E − 0 (–)0.30E − 0 (–)
    $ 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
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  • 收稿日期:  2021-07-12
  • 修回日期:  2021-08-22
  • 网络出版日期:  2022-06-01

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