A SCALE-INVARIANT HIGH-ORDER WCNS SCHEME
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摘要: 高速流场的数值模拟中, 既要保证对小尺度结构的高保真分辨, 又要实现对激波稳定、无振荡地捕捉.当前工程中广泛应用的高精度数值格式虽然都能一定程度地满足上述两种要求, 但仍与理想目标存在较大差距.例如, 模拟雷诺应力模型等小尺度问题时, 高精度格式在间断解附近易产生数值振荡.基于高精度格式所存在的上述问题, 本文引入去尺度函数, 探索了一种更加简单稳定的非线性权重构造方法, 并将其应用于7阶精度加权紧致非线性格式WCNS, 提出了一种尺度无关的7阶WCNS格式.该格式的性能与灵敏度参数和尺度因子的选择无关, 并且在小尺度下仍可以有效捕捉流场激波.同时, 该格式在间断处具有基本无振荡性质, 且在任意尺度函数下保持尺度无关, 并且在极值点处也能保持最优精度.本文还推导了7阶D权函数的形式.最后, 在一维线性对流方程中验证了新格式在流场光滑区能够达到设计精度, 并通过一系列数值实验证明了尺度无关的7阶WCNS格式在激波捕捉能力上具有良好表现, 为WCNS格式改进和解决可压缩湍流等非线性问题提供了一种新途径.Abstract: For numerical simulation of high-speed flows, it requires that small-scale structures are resolved with high-fidelity, and discontinuities are stably captured without spurious oscillation. These two aspects put forward almost contradictory requirements for numerical schemes. The widely used high-order schemes can satisfy the two demands required above to some extent. However, they all have advantages and disadvantages compared to each other and no one can be considered perfect. For example, high-order schemes are prone to generating numerical oscillations near discontinuities when a small-scale problem is discretized, such as the Reynolds-stress model. To solve this deficiency, a simple, effective and robust modification is introduced to the seventh-order weight compact nonlinear scheme (WCNS) by making use of the descaling function to formulate a scale-invariant WCNS scheme. The descaling function is devised using an average of the function values and introduced into the nonlinear weights of the WCNS7-JS/Z/D schemes to eliminate the scale dependency. The design idea of the scale-invariant WCNS scheme is to make weights independent of the scale factor and the sensitivity parameter. In addition, the shock-capturing ability of the new scheme performs well even for small-scale problems. The new schemes can achieve an essentially non-oscillatory approximation of a discontinuous function (ENO-property), a scale-invariant property with an arbitrary scale of a function (Si-property), and an optimal order of accuracy with smooth function regardless of the critical point (Cp-property). We derive the seventh-order D-type weights. The one-dimensional linear advection equation is solved to verify that WCNS schemes can achieve the optimal (seventh) order of accuracy. We test a series of one- and two-dimensional numerical experiments governed by Euler equations to demonstrate that the scale-invariant WCNS schemes perform well in the shock-capturing ability. Overall, the scale-invariant WCNS schemes provide a new method for improving WCNS schemes and solving nonlinear problems.
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Key words:
- WCNS scheme /
- descaling function /
- scale-invariant /
- shock-capturing
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图 1
$\varepsilon = 1.0 \times {10^{ - 6}}$ 时Lax问题的密度分布曲线Figure 1. Density distribution curves of the Lax problem with
$\varepsilon $ =$1.0 \times {10^{ - 6}} $ 
图 2
$\varepsilon = 1.0 \times {10^{ - 40}}$ 时Lax问题的密度分布曲线Figure 2. Density distribution curves of the Lax problem with
$\varepsilon $ =$ 1.0 \times {10^{ - 40}} $ 
图 3
$\varepsilon = 1.0 \times {10^{ - 6}}$ 和$\varepsilon = 1.0 \times {10^{ - 40}}$ 时使用WCNS7-Dm格式求解Lax问题的密度分布曲线Figure 3. Density distribution curves of the Lax problem computed by the WCNS7-Dm scheme with
$\varepsilon = 1.0 \times {10^{ - 6}}$ and$\varepsilon = 1.0 \times {10^{ - 40}}$ 
图 4 3套网格单元数
$N$ 下使用WCNS格式计算Sine函数$ {f_1}(x) $ 的Si误差${E_{Si}}(\lambda )$ ,$\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{-19}},1.0 \times {\text{1}}{{\text{0}}^{19}}]$ ,$\varepsilon = 1.0 \times {10^{ - 6}}$ (上行)和$\varepsilon =1.0 \times {10^{ - 40}}$ (下行)Figure 4. The Si-error
${E_{Si}}(\lambda )$ of the Sine function$ {f_1}(x) $ calculated by the WCNS schemes with$\varepsilon = 1.0 \times {10^{ - 6}}$ (top row),$\varepsilon = 1.0 \times {10^{ - 40}}$ (bottom row), and different scale factors$\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{ - 19}},1.0 \times {\text{1}}{{\text{0}}^{19}}]$ for three number of cells$N$ 
图 5 3套网格单元数
$N$ 下使用WCNS格式计算$ {\text{ReLU}} $ 函数$ {f_2}(x) $ 的Si误差${E_{Si}}(\lambda )$ ,$\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{-19}},1.0 \times{\text{1}}{{\text{0}}^{19}}]$ ,$\varepsilon = 1.0 \times{10^{ - 6}}$ (上行)和$\varepsilon = 1.0 \times{10^{ - 40}}$ (下行)Figure 5. The Si-error
${E_{Si}}(\lambda )$ of the$ {\text{ReLU}} $ function$ {f_2}(x) $ calculated by the WCNS schemes with$\varepsilon = 1.0 \times{10^{ - 6}}$ (top row),$\varepsilon =1.0 \times {10^{ - 40}}$ (bottom row), and different scale factors$\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{-19}},1.0 \times{\text{1}}{{\text{0}}^{19}}]$ for three number of cells$N$ 
图 6 3套网格单元数
$N$ 下使用WCNS格式计算Heaviside函数$ {f_3}(x) $ 的Si误差${E_{Si}}(\lambda )$ ,$\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{ - 19}},1.0 \times{\text{1}}{{\text{0}}^{19}}]$ ,$\varepsilon = 1.0 \times{10^{ - 6}}$ (上行)和$\varepsilon = 1.0 \times{10^{ - 40}}$ (下行)Figure 6. The Si-error
${E_{Si}}(\lambda )$ of the Heaviside function$ {f_3}(x) $ calculated by the WCNS schemes with$\varepsilon = 1.0 \times{10^{ - 6}}$ (top row),$\varepsilon = 1.0 \times{10^{ - 40}}$ (bottom row), and different scale factors$\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{ - 19}},1.0 \times{\text{1}}{{\text{0}}^{19}}]$ for three number of cells$N$ 
图 7 在极值点
${n_{cp}} = n = 1,2,3,4$ 下WCNS格式的收敛精度和${L_\infty }$ 误差, 且$\lambda = 1$ ,$\varepsilon = 1.0 \times {10^{ - 6}}$ (上行),$\varepsilon = 1.0 \times {10^{ - 40}}$ (下行)Figure 7.
${L_\infty }$ error of the WCNS schemes with$\varepsilon = 1.0 \times {10^{ - 6}}$ (top row),$\varepsilon = 1.0 \times {10^{ - 40}}$ (bottom row), and the scale factor$\lambda = 1$ for${n_{cp}} = n = 1,2,3,4$ 
图 8
$\varepsilon = 1.0 \times {10^{ - 6}}$ 和$\varepsilon = 1.0 \times {10^{ - 40}}$ 时分别使用WCNS格式计算$\lambda = 1.0 \times {10^{ - 7}},1,{100}$ 下的Lax问题Figure 8. Density of the Lax problem computed by the WCNS schemes with
$\varepsilon = 1.0 \times {10^{ - 6}}$ and$\varepsilon =1.0 \times {10^{ - 40}}$ for$\lambda = 1.0 \times {10^{ - 7}},1,{100}$ 
图 9
$\varepsilon =1.0 \times {10^{ - 40}}$ 时使用WCNS格式计算$ \lambda = 1.0 \times {10^{ - 7}} $ 的Sod问题Figure 9. Density of the Sod problem computed by the WCNS schemes with
$\varepsilon =1.0 \times {10^{ - 40}}$ for$ \lambda =1.0 \times {10^{ - 7}} $ 
图 10
$ \varepsilon =1.0 \times {10^{ - 40}} $ 时使用WCNS格式计算$\lambda =1.0 \times {10^{ - 7}}$ 激波密度波问题Figure 10. Density of the shock-density wave interaction problem computed by the WCNS schemes with
$ \varepsilon =1.0 \times {10^{ - 40}} $ for$\lambda $ =$1.0 \times {10^{ - 7}} $ 
图 11
$ N = 400 $ 时使用WCNS格式计算爆炸波相互作用问题Figure 11. Density of the blast-waves interaction problem computed by the WCNS schemes with
$ N = 400 $ 
图 12
$N = 800$ 时使用WCNS格式计算爆炸波相互作用问题Figure 12. Density of the blast-waves interaction problem computed by the WCNS schemes with
$ N = 800 $ 
图 13
$\varepsilon = 1.0 \times {10^{ - 40}}$ 时使用WCNS格式计算Riemann问题中的构型3Figure 13. Density of the configuration 3 of 2-D Riemann problems computed by the WCNS7-JSm/Zm/Dm schemes with
$ \varepsilon = 1.0 \times {10^{ - 40}} $ 
13
$\varepsilon =1.0 \times {10^{ - 40}}$ 时使用WCNS格式计算Riemann问题中的构型3 (续)13. Density of the configuration 3 of 2-D Riemann problems computed by the WCNS7-JSm/Zm/Dm schemes with
$ \varepsilon =1.0 \times {10^{ - 40}} $ (continued)
图 14
$\varepsilon = 1.0 \times {10^{ - 40}}$ 时使用WCNS7-D/Dm格式计算$\lambda = 1.0 \times {10^{ - 7}}$ 下的二维 Riemann 问题中的构型6Figure 14. Density of the configuration 6 of 2-D Riemann problems computed by the WCNS7-D/Dm schemes with
$\lambda = 1.0 \times {10^{ - 7}}$ and$\varepsilon = 1.0 \times {10^{ - 40}}$ 
14
$\varepsilon = 1.0 \times {10^{ - 40}}$ 时使用WCNS7-D/Dm格式计算$\lambda = 1.0 \times {10^{ - 7}}$ 下的二维 Riemann 问题中的构型6 (续)14. Density of the configuration 6 of 2-D Riemann problems computed by the WCNS7-D/Dm schemes with
$\lambda =1.0 \times {10^{ - 7}}$ and$\varepsilon = 1.0 \times {10^{ - 40}}$ (continued)
图 15
$\varepsilon =1.0 \times {10^{ - 40}}$ 时使用WCNS7-Dm格式计算双马赫反射问题Figure 15. Density of the double Mach reflection problem computed by the WCNS7-Dm scheme with
$\varepsilon =1.0 \times {10^{ - 40}}$ 
图 16
$\varepsilon = 1.0 \times {10^{ - 40}}$ 时使用WCNS7-Dm格式计算Richtmyer-Meshko不稳定性问题Figure 16. Density of the Richtmyer-Meshkov instability problem computed by the WCNS7-Dm scheme with
$\varepsilon = 1.0 \times {10^{ - 40}}$ 表 1 WCNS格式在
$\varepsilon = 1.0 \times 1{{\text{0}}^{ - 40}}$ ,${n_{cp}} = 3$ 时的${L_\infty }$ 误差${E_N}$ 和收敛阶$O(\Delta {x^s})$ Table 1.
${L_\infty }$ error${E_N}$ , and order of accuracy$O(\Delta {x^s})$ of WCNS schemes with$\varepsilon = 1.0 \times 1{{\text{0}}^{ - 40}}$ and${n_{cp}} = 3$ $\lambda $ $N$ WCNS7-JSm WCNS7-Zm WCNS7-D WCNS7-Dm ${E_N}$ $s$ ${E_N}$ $s$ ${E_N}$ $s$ ${E_N}$ $s$ $1.0 \times {10^{ - 7}}$ $ 80 $ $2.51 \times {10^{ - 12}}$ — $ 2.28 \times {10^{ - 12}} $ — $ 2.84 \times {10^{ - 18}} $ — $ 1.70 \times {10^{ - 15}} $ — $ 160 $ $ 3.09 \times {10^{ - 13}} $ $ 3.02 $ $ 2.74 \times {10^{ - 13}} $ $ 3.05 $ $ 2.24 \times {10^{ - 20}} $ $ 6.99 $ $ 2.00 \times {10^{ - 17}} $ $ 6.41 $ $ 320 $ $ 3.83 \times {10^{ - 14}} $ $ 3.01 $ $ 3.38 \times {10^{ - 14}} $ $ 3.02 $ $ 1.75 \times {10^{ - 22}} $ $ 6.99 $ $ 1.83 \times {10^{ - 19}} $ $ 6.77 $ $ 640 $ $ 4.76 \times {10^{ - 15}} $ $ 3.01 $ $ 4.19 \times {10^{ - 15}} $ $ 3.01 $ $ 1.37 \times {10^{ - 24}} $ $ 7.00 $ $ 1.52 \times {10^{ - 21}} $ $ 6.91 $ $ 1280 $ $ 5.93 \times {10^{ - 16}} $ $ 3.00 $ $ 5.21 \times {10^{ - 16}} $ $ 3.01 $ $ 1.07 \times {10^{ - 26}} $ $ 7.00 $ $ 1.21 \times {10^{ - 23}} $ $ 6.97 $ $ 2560 $ $ 7.40 \times {10^{ - 17}} $ $ 3.00 $ $ 6.50 \times {10^{ - 17}} $ $ 3.00 $ $ 8.39 \times {10^{ - 29}} $ $ 7.00 $ $ 9.57 \times {10^{ - 26}} $ $ 6.99 $ $ 5120 $ $ 9.24 \times {10^{ - 18}} $ $ 3.00 $ $ 8.12 \times {10^{ - 18}} $ $ 3.00 $ $ 6.56 \times {10^{ - 31}} $ — $ 7.51 \times {10^{ - 28}} $ $ 6.99 $ $10\;240$ $ 1.15 \times {10^{ - 18}} $ $ 3.00 $ $ 1.01 \times {10^{ - 18}} $ $ 3.00 $ $ 5.13 \times {10^{ - 33}} $ $ 7.00 $ $ 5.88 \times {10^{ - 30}} $ $ 7.00 $ ${100}$ $ 80 $ $ 2.51 \times {10^{ - 3}} $ — $ 2.28 \times {10^{ - 3}} $ — $ 7.54 \times {10^{ - 5}} $ — $ 1.70 \times {10^{ - 6}} $ — $ 160 $ $ 3.09 \times {10^{ - 4}} $ $ 3.02 $ $ 2.74 \times {10^{ - 4}} $ $ 3.05 $ $ 7.15 \times {10^{ - 7}} $ $ 6.72 $ $ 2.00 \times {10^{ - 8}} $ $ 6.41 $ $ 320 $ $ 3.83 \times {10^{ - 5}} $ $ 3.01 $ $ 3.38 \times {10^{ - 5}} $ $ 3.02 $ $ 5.42 \times {10^{ - 9}} $ $ 7.04 $ $ 1.83 \times {10^{ - 10}} $ $ 6.77 $ $ 640 $ $ 4.76 \times {10^{ - 6}} $ $ 3.01 $ $ 4.19 \times {10^{ - 6}} $ $ 3.01 $ $ 4.06 \times {10^{ - 11}} $ $ 7.06 $ $ 1.52 \times {10^{ - 12}} $ $ 6.91 $ $ 1280 $ $ 5.93 \times {10^{ - 7}} $ $ 3.00 $ $ 5.21 \times {10^{ - 7}} $ $ 3.01 $ $ 3.09 \times {10^{ - 13}} $ $ 7.04 $ $ 1.21 \times {10^{ - 14}} $ $ 6.97 $ $ 2560 $ $ 7.40 \times {10^{ - 8}} $ $ 3.00 $ $ 6.50 \times {10^{ - 8}} $ $ 3.00 $ $ 2.37 \times {10^{ - 15}} $ $ 7.02 $ $ 9.57 \times {10^{ - 17}} $ $ 6.99 $ $ 5120 $ $ 9.24 \times {10^{ - 9}} $ $ 3.00 $ $ 8.12 \times {10^{ - 9}} $ $ 3.00 $ $ 1.84 \times {10^{ - 17}} $ $ 7.01 $ $ 7.51 \times {10^{ - 19}} $ $ 6.99 $ $10\;240$ $ 1.15 \times {10^{ - 9}} $ $ 3.00 $ $ 1.01 \times {10^{ - 9}} $ $ 3.00 $ $ 1.43 \times {10^{ - 19}} $ $ 7.01 $ $ 5.88 \times {10^{ - 21}} $ $ 7.00 $ 
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表 2 WCNS7-D/Dm格式的误差值和收敛阶 (
$\varepsilon $ = 1.0 × 10−40)Table 2. Error and accuracy statistics for WCNS7-D/Dm schemes (
$\varepsilon $ = 1.0 × 10−40)$\lambda $ $N$ WCNS7-D WCNS7-Dm ${L_1}$error order ${L_\infty }$error order ${L_1}$error order ${L_\infty }$error order $1.0 \times {10^{ - 7}}$ $ 5.73 \times {10^{ - 9}} $ — $ 3.25 \times {10^{ - 8}} $ — $ 5.89 \times {10^{ - 9}} $ — $ 5.04 \times {10^{ - 8}} $ — $ 40 $ $ 1.27 \times {10^{ - 9}} $ $ 2.17 $ $ 7.92 \times {10^{ - 9}} $ $ 2.04 $ $ 1.09 \times {10^{ - 9}} $ $ 2.43 $ $ 1.10 \times {10^{ - 8}} $ $ 2.19 $ $ 80 $ $ 4.12 \times {10^{ - 11}} $ $ 4.95 $ $ 3.70 \times {10^{ - 10}} $ $ 4.42 $ $ 4.17 \times {10^{ - 11}} $ $ 4.71 $ $ 3.72 \times {10^{ - 10}} $ $ 4.89 $ $ 160 $ $ 3.97 \times {10^{ - 13}} $ $ 6.70 $ $ 4.17 \times {10^{ - 12}} $ $ 6.47 $ $ 3.97 \times {10^{ - 13}} $ $ 6.71 $ $ 4.17 \times {10^{ - 12}} $ $ 6.48 $ $ 320 $ $ 3.26 \times {10^{ - 15}} $ $ 6.93 $ $ 3.47 \times {10^{ - 14}} $ $ 6.91 $ $ 3.26 \times {10^{ - 15}} $ $ 6.93 $ $ 3.47\times {10^{ - 14}} $ $ 6.93 $ $ 640 $ $ 2.62 \times {10^{ - 17}} $ $ 6.96 $ $ 2.85 \times {10^{ - 16}} $ $ 6.93 $ $ 2.62 \times {10^{ - 17}} $ $ 6.96 $ $ 2.85 \times {10^{ - 16}} $ $ 6.93 $ 1 $ 20 $ $ 5.51 \times {10^{ - 2}} $ — $ 4.85 \times {10^{ - 1}} $ — $ 5.89 \times {10^{ - 2}} $ — $ 5.04 \times {10^{ - 1}} $ — $ 40 $ $ 8.75 \times {10^{ - 3}} $ $ 2.65 $ $ 9.15 \times {10^{ - 2}} $ $ 2.40 $ $ 1.09 \times {10^{ - 2}} $ $ 2.43 $ $ 1.10 \times {10^{ - 1}} $ $ 2.19 $ $ 80 $ $ 4.07 \times {10^{ - 4}} $ $ 4.43 $ $ 3.70 \times {10^{ - 3}} $ $ 4.63 $ $ 4.16 \times {10^{ - 4}} $ $ 4.17 $ $ 3.72 \times {10^{ - 3}} $ $ 4.89 $ $ 160 $ $ 3.97 \times {10^{ - 6}} $ $ 6.68 $ $ 4.17 \times {10^{ - 5}} $ $ 6.47 $ $ 3.97 \times {10^{ - 6}} $ $ 6.71 $ $ 4.17 \times {10^{ - 5}} $ $ 6.48 $ $ 320 $ $ 3.26 \times {10^{ - 8}} $ $ 6.93 $ $ 3.47 \times {10^{ - 7}} $ $ 6.91 $ $ 3.26 \times {10^{ - 8}} $ $ 6.93 $ $ 3.47 \times {10^{ - 7}} $ $ 6.91 $ $ 640 $ $ 2.62 \times {10^{ - 10}} $ $ 6.96 $ $ 2.85 \times {10^{ - 9}} $ $ 6.93 $ $ 2.62 \times {10^{ - 10}} $ $ 6.96 $ $ 2.85 \times {10^{ - 9}} $ $ 6.93 $ ${10^2}$ $ 20 $ $ 5.89 \times {10^0} $ — $ 5.04 \times {10^{ - 1}} $ — $ 5.89 \times {10^0} $ — $ 5.04 \times {10^{ - 1}} $ — $ 40 $ $ 1.10 \times {10^0} $ $ 2.42 $ $ 1.11 \times {10^{ - 1}} $ $ 2.18 $ $ 1.09 \times {10^0} $ $ 2.43 $ $ 1.10 \times {10^{ - 1}} $ $ 2.19 $ $ 80 $ $ 4.43 \times {10^{ - 2}} $ $4.63$ $ 3.71 \times {10^{ - 1}} $ $ 4.90 $ $ 4.16 \times {10^{ - 2}} $ $ 4.71 $ $ 3.72 \times {10^{ - 1}} $ $ 4.89 $ $ 160 $ $ 3.97 \times {10^{ - 4}} $ $ 6.80 $ $ 4.17 \times {10^{ - 3}} $ $ 6.47 $ $ 3.97 \times {10^{ - 4}} $ $ 6.71 $ $ 4.17 \times {10^{ - 3}} $ $ 6.48 $ $ 320 $ $ 3.26 \times {10^{ - 6}} $ $ 6.93 $ $ 3.47 \times {10^{ - 5}} $ $ 6.91 $ $ 3.26 \times {10^{ - 6}} $ $ 6.93 $ $ 3.47 \times {10^{ - 5}} $ $ 6.91 $ $ 640 $ $ 2.62 \times {10^{ - 8}} $ $ 6.96 $ $ 2.85 \times {10^{ - 7}} $ $ 6.93 $ $ 2.62 \times {10^{ - 8}} $ $ 6.96 $ $ 2.85 \times {10^{ - 7}} $ $ 6.93 $
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