INVESTIGATION OF THE TIME EFFICIENCY OF THE SEVENTH-ORDER WENO-S SCHEME
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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了数值模拟. 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.Abstract: The WENO-S scheme is a class of weighted essentially non-oscillatory schemes suitable for numerical simulations of problems with discontinuities. The smoothness indicator of this kind of scheme is constant for single-frequency waves, which makes this kind of scheme have exactly the same approximate dispersion relationship with its linear base scheme, and thus has an excellent ability to simulate small-scale waves. Time efficiency is crucial for numerical methods. For a WENO-S scheme, the formula of the smoothness indicator on each sub-stencil has the same formula except for different subscripts. Then some smoothness indicators are the same when calculating adjacent numerical fluxes of linear convection equations. So, a method is proposed to remove redundant computations of smoothness indicators. The premise of this approach is that the quantity used for reconstruction or interpolation on a grid line can be represented as a sequence. According to this requirement, the feasibility and application requirements for several different physical problems are analyzed. The seventh-order WENO-S scheme is employed to illustrate the advantages of the WENO-S schemes, including good properties near extreme points, good stability near discontinuities, and outstanding spectral properties. Then the method of eliminating the computation of the redundant smoothness indicators is introduced. In numerical computation, all smoothness indicators in a grid line are calculated and stored in advance. With this approach, the count of the smoothness indicator calculation is about 1/4 of the original one for the seventh-order WENO-S scheme when there are many grid points. Numerical examples include one-dimensional advection, spherical wave propagation, two-dimensional rotation, small disturbance propagation, and one- and two- dimensional inviscid flow problems. The numerical results show that this scheme can capture a variety of flow structures well and have good time efficiency. Furthermore, the proposed method reduces the computational time by about 20%.
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Key words:
- WENO scheme /
- smoothness indicator /
- shock-capturing scheme /
- high order scheme /
- time efficiency
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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-JS0 59(N + 1) 87(N + 1) 0 146(N + 1) WENO7-JS1 55(N + 1) 75(N + 1) 0 130(N + 1) WENO7-Z 62(N + 1) 75(N + 1) N + 1 138(N + 1) WENO7-S0 77(N + 1) 51(N + 1) 5(N + 1) 133(N + 1) WENO7-S1 47N + 77 39N + 51 2N + 5 88N + 133 
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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
Function WENO7-JS WENO7-Z WENO7-S ${f_1}\left( x \right)$ average 1.56 × 10−3 2.40 × 10−3 2.58 × 10−4 minimum 3.61 × 10−2 1.00 × 10−2 7.58 × 10−4 ${f_2}\left( x \right)$ average 8.49 × 10−4 9.31 × 10−6 4.38 × 10−8 minimum 2.40 × 10−3 4.42 × 10−5 7.27 × 10−8 ${f_3}\left( x \right)$ average 9.03 × 10−2 4.11 × 10−2 2.07 × 10−2 minimum 1.87 × 10−1 1.14 × 10−1 5.43 × 10−2
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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
Scheme WENO7-JS0 WENO7-JS1 WENO7-Z WENO7-S0 WENO7-S1 time/s 636.39 580.32 645.67 546.09 416.99
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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
Scheme WENO7-JS0 WENO7-JS1 WENO7-Z WENO7-S0 WENO7-S1 time/s 1.422 1.336 1.465 1.219 0.953
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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
Grid WENO7-JS WENO7-Z WENO7-S L1 error 200 × 200 1.759 1.428 1.600 400 × 400 0.959 0.788 0.886 800 × 800 0.524 0.430 0.484 up/down overshooting amplitude $\delta$ 200 × 200 1.11 × 10−4 1.45 × 10−2 5.02 × 10−3 400 × 400 9.86 × 10−5 9.13 × 10−3 2.72 × 10−4 800 × 800 1.27 × 10−4 2.58 × 10−2 2.87 × 10−8
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表 6 二维旋转问题的计算时间
Table 6. Computational time for the two-dimensional rotation problem
Scheme WENO7-JS0 WENO7-JS1 WENO7-Z WENO7-S0 WENO7-S1 time/s 114.731 107.400 117.493 99.744 76.327
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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
Scheme WENO7-JS0 WENO7-JS1 WENO7-Z WENO7-S0 WENO7-S1 time/s 8.842 7.780 8.377 7.485 6.150
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表 8 Shu-Osher问题计算时间, N = 2000
Table 8. Computational time of Shu-Osher problem, N = 2000
Scheme WENO7-JS0 WENO7-JS1 WENO7-Z WENO7-S0 WENO7-S1 time/s 3.330 3.037 3.317 3.015 2.255
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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
Scheme WENO7-JS0 WENO7-JS1 WENO7-Z WENO7-S0 WENO7-S1 time/s 14.350 13.195 14.520 13.007 10.159
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