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7阶WENO-S格式的计算效率研究

武从海 李虎 刘旭亮 罗勇 张树海

武从海, 李虎, 刘旭亮, 罗勇, 张树海. 7阶WENO-S格式的计算效率研究. 力学学报, 2023, 55(1): 1-15 doi: 10.6052/0459-1879-22-371
引用本文: 武从海, 李虎, 刘旭亮, 罗勇, 张树海. 7阶WENO-S格式的计算效率研究. 力学学报, 2023, 55(1): 1-15 doi: 10.6052/0459-1879-22-371
Wu Conghai, Li Hu, Liu Xuliang, Luo Yong, Zhang Shuhai. Investigation of the time efficiency of the seventh-order WENO-S scheme. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 1-15 doi: 10.6052/0459-1879-22-371
Citation: Wu Conghai, Li Hu, Liu Xuliang, Luo Yong, Zhang Shuhai. Investigation of the time efficiency of the seventh-order WENO-S scheme. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 1-15 doi: 10.6052/0459-1879-22-371

7阶WENO-S格式的计算效率研究

doi: 10.6052/0459-1879-22-371
基金项目: 国家自然科学基金(12172374, 12102450, 11732016)和国家数值风洞工程资助项目
详细信息
    作者简介:

    武从海, 副研究员, 主要研究方向: 计算流体力学高精度方法及计算气动声学. E-mail: wraiment@163.com

  • 中图分类号: V211.3, O241.8

INVESTIGATION OF THE TIME EFFICIENCY OF THE SEVENTH-ORDER WENO-S SCHEME

  • 摘要: WENO-S格式是一类适合于含间断问题数值模拟的加权本质无振荡格式. 这类格式的光滑因子满足对单频波为常数, 这使得其近似色散关系与线性基底格式一致, 并且具有良好的小尺度波动模拟能力.计算效率是数值方法性能指标的一个重要方面. 由于WENO-S格式的光滑因子在各子模板上的计算公式除下标不同外形式一致, 在计算线性对流方程相邻数值通量时, 部分光滑因子完全相同.为此提出一种消除WENO-S格式冗余光滑因子计算的方法. 该方法要求一条网格线上用于重构或插值的量可以表示为一个序列. 基于此要求分析其对于几种不同物理问题的可行性和使用方法. 以7阶WENO-S格式为例介绍了格式性质和去除冗余光滑因子计算的方法. 该方法中预先计算和存储一条网格线上的所有光滑因子, 在网格点较多的情况下, 光滑因子计算次数约为原7阶WENO-S格式的1/4. 对一维对流问题、球面波传播问题、二维旋转问题、二维小扰动传播问题及一维和二维无黏流动问题进行数值模拟. 结果表明该格式对多种流动结构具有良好的捕捉能力, 并且同时具有良好的计算效率, 去除冗余计算后又降低了约20%的计算时间.

     

  • 图  1  圆周和正弦函数的光滑程度示意图

    Figure  1.  Schematic diagram of the smoothness of a circle and a sine function

    图  2  组合波的远距传输问题, 网格点数N = 400, 计算终止时间T = 2000

    Figure  2.  Long-distance advection of combined waves with the number of grid points N = 400 and the end time T = 2000

    图  3  球面波传播问题, 计算终止时间为T = 400, 空间步长为dr = 1. 时间步长为dt = 0.1

    Figure  3.  The spherical wave propagation problem with the end time T = 400, the grid length dr = 1, and the time step dt = 0.1

    图  4  二维旋转问题计算结果与精确解

    Figure  4.  The computational results and the exact solution of the two-dimensional rotation problem

    图  5  二维旋转问题两条线上的结果对比图

    Figure  5.  Comparison of results on two lines of the two-dimensional rotation problem

    图  6  二维小扰动传播问题计算结果, 网格200 × 200, 终止时间T = 30

    Figure  6.  Computational results of two-dimensional small disturbance propagation problem, with grid 200 × 200 and end time T = 30

    图  7  Shu-Osher 问题计算结果的密度对比

    Figure  7.  The density distributions of computational results of Shu-Osher problem

    图  8  激波旋涡相互作用问题, 压强等值线范围1.19-1.37, 共90条, 网格200 × 100, 终止时间T = 0.6

    Figure  8.  Shock vortex interaction problem. 90 pressure isolines ranging from 1.19 to 1.37. Component-wise seventh-order WENO schemes with grid 200 × 100 and end time T = 0.6

    表  1  几种WENO格式计算N + 1个数值通量所需浮点运算量

    Table  1.   The number of floating point operations required to calculate N + 1 numerical fluxes for several WENO schemes

    Scheme +, −*, /|·|Total
    WENO7-JS059(N + 1)87(N + 1)0146(N + 1)
    WENO7-JS155(N + 1)75(N + 1)0130(N + 1)
    WENO7-Z62(N + 1)75(N + 1)N + 1138(N + 1)
    WENO7-S077(N + 1)51(N + 1)5(N + 1)133(N + 1)
    WENO7-S147N + 7739N + 512N + 588N + 133
    下载: 导出CSV

    表  2  权系数偏离值的平均值与最大值

    Table  2.   The average values and maximum values of the deviations of weights

    FunctionWENO7-JSWENO7-ZWENO7-S
    ${f_1}\left( x \right)$average1.56 × 10−32.40 × 10−32.58 × 10−4
    minimum3.61 × 10−21.00 × 10−27.58 × 10−4
    ${f_2}\left( x \right)$average8.49 × 10−49.31 × 10−64.38 × 10−8
    minimum2.40 × 10−34.42 × 10−57.27 × 10−8
    ${f_3}\left( x \right)$average9.03 × 10−24.11 × 10−22.07 × 10−2
    minimum1.87 × 10−11.14 × 10−15.43 × 10−2
    下载: 导出CSV

    表  3  组合波远距传输问题的计算时间

    Table  3.   Computational time of long-distance advection of combined waves

    SchemeWENO7-JS0WENO7-JS1WENO7-ZWENO7-S0WENO7-S1
    time/s636.39580.32645.67546.09416.99
    下载: 导出CSV

    表  4  球面波传播问题的计算时间, dt = 0.01

    Table  4.   Computational time of spherical wave propagation problem with dt = 0.01

    SchemeWENO7-JS0WENO7-JS1WENO7-ZWENO7-S0WENO7-S1
    time/s1.4221.3361.4651.2190.953
    下载: 导出CSV

    表  5  几种格式不同网格下计算结果的L1误差和上冲/下冲幅度

    Table  5.   The L1 error and up/down overshooting amplitude of the computational results with different schemes and grids

    GridWENO7-JSWENO7-ZWENO7-S
    L1 error200 × 2001.7591.4281.600
    400 × 4000.9590.7880.886
    800 × 8000.5240.4300.484
    up/down overshooting amplitude $\delta$200 × 2001.11 × 10−41.45 × 10−25.02 × 10−3
    400 × 4009.86 × 10−59.13 × 10−32.72 × 10−4
    800 × 8001.27 × 10−42.58 × 10−22.87 × 10−8
    下载: 导出CSV

    表  6  二维旋转问题的计算时间

    Table  6.   Computational time for the two-dimensional rotation problem

    SchemeWENO7-JS0WENO7-JS1WENO7-ZWENO7-S0WENO7-S1
    time/s114.731107.400117.49399.74476.327
    下载: 导出CSV

    表  7  二维小扰动传播问题的计算时间

    Table  7.   Computational time of two-dimensional small disturbance propagation problem

    SchemeWENO7-JS0WENO7-JS1WENO7-ZWENO7-S0WENO7-S1
    time/s8.8427.7808.3777.4856.150
    下载: 导出CSV

    表  8  Shu-Osher问题计算时间, N = 2000

    Table  8.   Computational time of Shu-Osher problem, N = 2000

    SchemeWENO7-JS0WENO7-JS1WENO7-ZWENO7-S0WENO7-S1
    time/s3.3303.0373.3173.0152.255
    下载: 导出CSV

    表  9  激波旋涡相互作用问题的计算时间

    Table  9.   Computational time of shock vortex interaction problem

    SchemeWENO7-JS0WENO7-JS1WENO7-ZWENO7-S0WENO7-S1
    time/s14.35013.19514.52013.00710.159
    下载: 导出CSV
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  • 收稿日期:  2022-08-15
  • 录用日期:  2022-11-11
  • 网络出版日期:  2022-11-12

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