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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,2022,43(2):224-236 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,2022,43(2):224-236 doi: 10.21656/1000-0887.420091
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. doi: 10.21656/1000-0887.420091
Citation: 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. 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);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

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

  • 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。

  • 图  1  三阶WENO数值通量的模板

    Figure  1.  Stencils for the 3rd-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 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 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 linear advection eq. (31) with initial value (35), at $t=6, \;N=400$

    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$

    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$CCFL = 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$

    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$

    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

    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 linear advection eq. (31) with 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)
    $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 $$ 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-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)
    $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 $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-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)
    $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 $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-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)
    $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
  • [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
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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
  • 刊出日期:  2022-02-01

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