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求解双曲守恒律方程的三阶修正模板WENO格式

王亚辉

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,2021,43(X):1-14 doi: 10.21656/1000-0887.420091
引用本文: 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,2021,43(X):1-14 doi: 10.21656/1000-0887.420091
Wang Yahui. Third-Order Modified Stencil WENO Schemes for Hyperbolic Conservation Laws[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420091
Citation: Wang Yahui. Third-Order Modified Stencil WENO Schemes for Hyperbolic Conservation Laws[J]. Applied Mathematics and Mechanics. 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
基金项目: 国家自然科学基金(12071470)
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

Third-Order Modified Stencil WENO Schemes for Hyperbolic Conservation Laws

  • 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$\varphi(x)$,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。
  • 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(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 the linear advection eq. (31)with initial value(33) at $t=40$

    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 the linear advection eq. (31) with initial value(34) at $t=41$$N=400$

    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 the linear advection eq. (31) with initial value(35) at $t=6$ with 400 grid points

    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 Sod problem[16] at $t=0.2$$N=200$$C_{\rm{CFL}}=0.6$

    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_{\rm{CFL}}=0.6$

    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 $C_{\rm{CFL}}=0.6$

    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$。(a)WENO-JS3 (b)WENO-Z3 (c)WENO-MS-JS3 (d)WENO-MS-Z3。

    Figure  8.  Density contours of the Rayleigh-Taylor instability at $t=1.95$ using $\Delta x=\Delta y=1/240$$C_{\rm{CFL}}=0.6$。(a)WENO-JS3 (b)WENO-Z3 (c) WENO-MS-JS3 (d) WENO-MS-Z3。

    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 double Mach reflection problem at $t=0.2$ with $1\;920\times480$ grid points。

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

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

    $ N $WENO-JS3WENO-Z3WENO-MS-JS3WENO-MS-Z3
    $ 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)
    $ 1280 $2.43E$ - $5(2.34)5.55E$ - $6(2.83)1.78E$ - $6(3.02)1.05E$ - $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 $$ L^{\infty} $误差和收敛阶

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

    $ N $WENO-JS3WENO-Z3WENO-MS-JS3WENO-MS-Z3
    $ 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)
    $ 1280 $6.38E$ - $4(1.58)2.25E$ - $4(1.99)5.47E$ - $5(3.00)3.35E$ - $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 $L1误差和收敛阶

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

    $ N $WENO-JS3WENO-Z3WENO-MS-JS3WENO-MS-Z3
    $ 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)
    $ 1280 $3.02E$ - $5(2.28)7.12E$ - $6(2.78)2.36E$ - $6(3.02)1.37E$ - $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 $的误差和收敛阶

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

    $ N $WENO-JS3WENO-Z3WENO-MS-JS3WENO-MS-Z3
    $ 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)
    $ 1280 $7.82E$ - $4(1.56)2.88E$ - $4(1.93)7.87E$ - $5(2.83)4.76E$ - $5(2.93)
    下载: 导出CSV
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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

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