﻿ CFD中统计误差的数值精度分析
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CFD中统计误差的数值精度分析

Accuracy analysis of numerical error with statistical forms in CFD
MIN Yaobing, MA Yankai, LI Song
China Aerodynamics Research and Development Center, Mianyang 621000, China
Abstract: Generally, the accuracy of an algorithm should be validated numerically in Computational Fluid Dynamics(CFD), and the research object is composed by statistical norms of numerical error, which is usually represented by the L1 norm, the L2 norm, and the L norm based on the hypothesis that each norm is equivalent in accuracy order. In reality, there are locally discontinuities of flow variables, non-smoothness of mesh, and nonlinear interpolations near critical points that will result in the loss in accuracy of numerical algorithm, causing different numerical orders of accuracy of each norm of numerical error. By carrying out detailed theoretical analysis, the relationship of different error norms in accuracy order is exhibited in this paper and is soon validated by numerical experiments. The research results in this paper can not only serve as a guide to the validation of the accuracy order in a CFD algorithm, but also theoretically support the judgement of numerical simulation order with more complex flow.
Keywords: statistical error    norm of error    numerical accuracy    analysis of accuracy    loss in accuracy

1 统计误差的数值精度分析

 ${e_i} = {c_i}{h^{{r_i}}}\;\;\;\;i = 1,2, \cdots ,{N_h}$ （1）

 $L_m^h \buildrel \Delta \over = {\left( {\frac{1}{{{N_h}}}\sum\limits_{i = 1}^{{N_h}} {{{\left| {{e_i}} \right|}^m}} } \right)^{\frac{1}{m}}}\;\;\;m \ge 1$ （2）

 ${\left\| {{\mathit{\boldsymbol{e}}_h}} \right\|_{{L_m}}} \buildrel \Delta \over = L_m^h$ （3）

 $\left\{ \begin{array}{l} L_1^h \buildrel \Delta \over = {\left\| {{\mathit{\boldsymbol{e}}_h}} \right\|_{{L_1}}} = \frac{1}{{{N_h}}}\sum\limits_{i = 1}^{{N_h}} {\left| {{e_i}} \right|} \\ L_2^h \buildrel \Delta \over = {\left\| {{\mathit{\boldsymbol{e}}_h}} \right\|_{{L_2}}} = \sqrt {\frac{1}{{{N_h}}}\sum\limits_{i = 1}^{{N_h}} {{{\left| {{e_i}} \right|}^2}} } \\ L_\infty ^h \buildrel \Delta \over = {\left\| {{\mathit{\boldsymbol{e}}_h}} \right\|_{{L_\infty }}} = \max \left( {\left| {{e_i}} \right|} \right) \end{array} \right.$ （4）

1.1 各点精度不完全一致的情形

 $\min\left( {{r_i}} \right) < \max\left( {{x_i}} \right)$ （5）

 ${R_1} = \min \left( {{r_i}} \right) < {R_2} < \cdots < {R_n} = \max \left( {{r_i}} \right)$ （6）

 $\sum\limits_{i = 1}^n {{N_i}} = {N_h}$ （7）

 $L_m^h = {\left( {\frac{1}{{{N_h}}}\sum\limits_{j = 1}^n {{h^{m{R_j}}}} \sum\limits_{i = 1}^{{N_j}} {{{\left| {{c_i}} \right|}^m}} } \right)^{\frac{1}{m}}}$ （8）

 ${C_j} = {\left( {\frac{1}{{{N_j}}}\sum\limits_{i = 1}^{{N_j}} {{{\left| {{c_i}} \right|}^m}} } \right)^{\frac{1}{m}}}\;\;\;j = 1,2, \cdots ,n$ （9）

 $L_m^h = {\left( {\frac{1}{{{N_h}}}\sum\limits_{j = 1}^n {{h^{m{R_j}}}} {N_j}C_j^m} \right)^{\frac{1}{m}}}$ （10）

 ${r_{{\rm{fc}}}} = \frac{{{h_{\rm{c}}}}}{{{h_{\rm{f}}}}}$ （11）

 ${N_{{h_{\rm{f}}}}} = r_{{\rm{f}}{{\rm{c}}^N}}^d \cdot {N_{{h_{\rm{c}}}}}$ （12）

 ${N_{j,{\rm{f}}}} = r_{{\rm{f}}{{\rm{c}}^j}}^d \cdot {N_{j,{\rm{c}}}}\;\;\;\;j = 1,2, \cdots ,n$ （13）

 ${d_j} \le {d_N}\;\;\;j = 1,2, \cdots ,n$ （14）

 ${N_h} = \frac{{N_h^0}}{{{h^{{d_N}}}}},{N_j} = \frac{{N_j^0}}{{{h^{{d_j}}}}}\;\;\;\;j = 1,2, \cdots ,n$ （15）

 $L_m^h = {\left( {{h^{{d_N}}}\sum\limits_{j = 1}^n {{h^{m{R_j} - {d_j}}}} \frac{{N_j^0}}{{N_h^0}}C_j^m} \right)^{\frac{1}{m}}}$ （16）

 ${{\bar R}_{j,m}} = {R_j} - \frac{{{d_j}}}{m}$ （17）

 $\begin{array}{l} L_m^h = {\left( {{h^{{d_N}}}\sum\limits_{j = 1}^n {{h^{m{{\bar R}_{j,m}}}}} \frac{{N_j^0}}{{N_h^0}}C_j^m} \right)^{\frac{1}{m}}} = \\ \;\;\;\;\;{h^{\min\left( {{{\bar R}_{j,m}}} \right) + \frac{{{d_N}}}{m}}}{\left( {\sum\limits_{j = 1}^n {{h^{m\left( {{{\bar R}_{j,m}} - \min\left( {{{\bar R}_{j,m}}} \right)} \right)}}} \frac{{N_j^0}}{{N_h^0}}C_j^m} \right)^{\frac{1}{m}}} \end{array}$ （18）

 ${{\bar R}_{j,m}} = \min \left( {{{\bar R}_{j,m}}} \right) \ge 0$ （19）

 ${O_m} = \text{lg}{_{\frac{{{h_{\rm{c}}}}}{{{h_{\rm{f}}}}}}}\frac{{L_m^{{h_{\rm{c}}}}}}{{L_m^{{h_{\rm{f}}}}}} = \frac{1}{{\text{ln}{r_{{\rm{fc}}}}}}\text{ln}\frac{{L_m^{{h_{\rm{c}}}}}}{{L_m^{{h_{\rm{f}}}}}}$ （20）

 $\begin{array}{l} {O_m} = \left( {\min\left( {{{\bar R}_{j,m}}} \right) + \frac{{{d_N}}}{m}} \right)\frac{1}{{\text{ln}{r_{{\rm{fc}}}}}}\text{ln}\frac{{{h_{\rm{c}}}}}{{{h_{\rm{f}}}}} + \\ \;\;\;\;\;\;\;\frac{1}{{m \cdot \text{ln}{r_{{\rm{fc}}}}}}\text{ln}\frac{{\sum\limits_{j = 1}^n {h_{\rm{c}}^{m\left( {{{\bar R}_{j,m}} - \min\left( {{{\bar R}_{j,m}}} \right)} \right)}} \frac{{N_j^0}}{{N_h^0}}C_j^m}}{{\sum\limits_{j = 1}^n {h_{\rm{f}}^{m\left( {{{\bar R}_{j,m}} - \min\left( {{{\bar R}_{j,m}}} \right)} \right)}} \frac{{N_j^0}}{{N_h^0}}C_j^m}} \end{array}$ （21）

 ${O_m} = \min \left( {{R_j} - \frac{{{d_j}}}{m}} \right) + \frac{{{d_N}}}{m}$ （22）

 ${O_m} = {R_1} + \frac{{{d_N} - {d_1}}}{m}$ （23）

m趋于正无穷大时，有

 ${O_\infty } = \mathop {\min}\limits_{m \to + \infty } \left( {{O_m}} \right) = \min\left( {{R_j}} \right) = {R_1}$ （24）

 ${O_1} \ge {O_2} \ge \cdots \ge {O_\infty }$ （25）

1.2 各点精度一致的情形

 $\min \left( {{r_i}} \right) = \max \left( {{r_i}} \right) = R$ （26）

 ${L_m} = {\left( {\frac{1}{{{N_h}}}{h^{mR}}{N_h}C_h^m} \right)^{\frac{1}{m}}} = {h^R}{C_h}$ （27）

 ${C_h} = {\left( {\frac{1}{{{N_h}}}\sum\limits_{i = 1}^{{N_h}} {{{\left| {{c_i}} \right|}^m}} } \right)^{\frac{1}{m}}}$ （28）

 ${O_m} = \frac{1}{{\text{ln}{r_{{\rm{fc}}}}}}\text{ln}\frac{{{L_{m,{\rm{c}}}}}}{{{L_{m,{\rm{f}}}}}} = \frac{R}{{\text{ln}{r_{{\rm{fc}}}}}}\text{ln}\frac{{{h_{\rm{c}}}}}{{{h_{\rm{f}}}}}$ （29）

 ${O_m} = R$ （30）

2 数值算例

2.1 网格生成

 $\begin{array}{l} {\rm{do}}\;\;\;\;j = 1,{N_j}\\ {\rm{do}}\;\;\;\;i = 1,{N_i}\\ \;\;\;\;\;\;\;\Delta x = \frac{1}{{{N_i} - 1}},\Delta y = \frac{1}{{{N_j} - 1}}\\ \;\;\;\;\;\;\;{x_0} = i \cdot \Delta x - \frac{1}{2},{y_0} = j \cdot \Delta y - \frac{1}{2}\\ \;\;\;\;\;\;\;{x_1} = {\rm{ \mathsf{ π} }}{y_0} + \frac{{\rm{ \mathsf{ π} }}}{2},{y_1} = {\sin ^2}\left( {{x_1}} \right)\\ \;\;\;\;\;\;\;{\rm{if}}\left( {\left| {{x_0}} \right| < 0.4} \right)\;\;\;\;{\rm{then}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;{x_2} = \frac{{5{\rm{ \mathsf{ π} }}}}{4}{x_2} + \frac{{\rm{ \mathsf{ π} }}}{2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;{y_2} = {\sin ^\kappa }\left( {{x_2}} \right)\\ \;\;\;\;\;\;\;{\rm{else}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;{y_2} = 0\\ \;\;\;\;\;\;\;{\rm{end}}\;{\rm{if}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;{x_{i,j}} = {x_0}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;{y_{i,j}} = {y_0} + A \cdot {y_1} \cdot {y_2}\\ \;\;\;\;\;\;\;{\rm{end}}\;{\rm{do}}\\ \;\;\;\;\;\;\;{\rm{end}}\;{\rm{do}} \end{array}$ （31）

 图 1 函数y2分布 Fig. 1 Distribution of function y2
2.2 数值离散

 ${I_x} = \frac{{\partial {{\hat \xi }_x}}}{{\partial \xi }} + \frac{{\partial {{\hat \eta }_x}}}{{\partial \eta }}$ （32）

 $\left\{ {\begin{array}{*{20}{l}} {{{\hat \xi }_x} = {J^{ - 1}}{\xi _x} = {y_\eta }}\\ {{{\hat \eta }_x} = {J^{ - 1}}{\eta _x} = - {y_\xi }} \end{array}} \right.$ （33）

 $I_x^N = {\rm{ \mathsf{ δ} }}_\xi ^1\left( {{{\hat \xi }_x}} \right) + \delta _\eta ^1\left( {{{\hat \eta }_x}} \right) = \delta _\xi ^1\left( {\delta _\eta ^2y} \right) - \delta _\eta ^1\left( {\delta _\xi ^2y} \right)$ （34）

 $\begin{array}{*{20}{c}} {{{\left( {{\delta _\xi }E} \right)}_j} = \frac{\alpha }{{2\Delta \xi }}\left( {{E_{j + 1}} - {E_{j - 1}}} \right) + }\\ {\frac{{1 - \alpha }}{{4\Delta \xi }}\left( {{E_{j + 2}} - {E_{j - 2}}} \right)} \end{array}$ （35）

 $\left\{ {\begin{array}{*{20}{l}} {{\delta ^1}:{\alpha _1} = \frac{1}{3}}\\ {{\delta ^2}:{\alpha _2} = \frac{2}{3}} \end{array}} \right.$ （36）
2.3 结果分析

 $\left\{ {\begin{array}{*{20}{l}} {\kappa = 1}&{{R_1} = 0}\\ {\kappa = 2}&{{R_1} = 1}\\ {\kappa = 3}&{{R_1} = 2} \end{array}} \right.$ （37）

κ=1时，几何守恒律误差IxN在整个计算区域上统计误差的Lm范数的数值精度Omκ=1

 $O_m^{\kappa = 1} = \frac{1}{m}$ （38）

 $O_m^{\kappa = 2} = 1 + \frac{1}{m}$ （39）

 $O_m^{\kappa = 3} = 2$ （40）

 N L1 L2 L3 L4 L5 L∞ 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 200 2.67E-03 2.04E-02 4.18E-02 6.03E-02 7.55E-02 2.06E-01 300 1.77E-03 1.01 1.67E-02 0.49 3.65E-02 0.33 5.46E-02 0.25 6.97E-02 0.20 2.06E-01 0 500 1.06E-03 1.01 1.30E-02 0.50 3.09E-02 0.33 4.81E-02 0.25 6.30E-02 0.20 2.06E-01 0 800 6.58E-04 1.01 1.03E-02 0.50 2.64E-02 0.33 4.28E-02 0.25 5.74E-02 0.20 2.06E-01 0 1 200 4.38E-04 1.00 8.39E-03 0.50 2.31E-02 0.33 3.86E-02 0.25 5.29E-02 0.20 2.06E-01 0 2 000 2.62E-04 1.00 6.50E-03 0.50 1.95E-02 0.33 3.40E-02 0.25 4.78E-02 0.20 2.06E-01 0 3 000 1.75E-04 1.00 5.31E-03 0.50 1.70E-02 0.33 3.07E-02 0.25 4.41E-02 0.20 2.06E-01 0 5 000 1.05E-04 1.00 4.11E-03 0.50 1.43E-02 0.33 2.71E-02 0.25 3.98E-02 0.20 2.06E-01 0 8 000 6.55E-05 1.00 3.25E-03 0.50 1.23E-02 0.33 2.41E-02 0.25 3.62E-02 0.20 2.06E-01 0

 N L1 L2 L3 L4 L5 L∞ 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 200 1.75E-04 7.03E-04 1.40E-03 2.05E-03 2.61E-03 8.07E-03 300 7.80E-05 1.99 3.82E-04 1.50 8.18E-04 1.33 1.24E-03 1.25 1.61E-03 1.20 5.38E-03 1.00 500 2.82E-05 1.99 1.77E-04 1.50 4.14E-04 1.33 6.54E-04 1.25 8.72E-04 1.20 3.23E-03 1.00 800 1.10E-05 2.00 8.75E-05 1.50 2.22E-04 1.33 3.64E-04 1.25 4.96E-04 1.20 2.02E-03 1.00 1 200 4.90E-06 2.00 4.76E-05 1.50 1.29E-04 1.33 2.19E-04 1.25 3.05E-04 1.20 1.35E-03 1.00 2 000 1.77E-06 2.00 2.21E-05 1.50 6.53E-05 1.33 1.16E-04 1.25 1.65E-04 1.20 8.07E-04 1.00 3 000 7.85E-07 2.00 1.20E-05 1.50 3.81E-05 1.33 6.97E-05 1.25 1.02E-04 1.20 5.38E-04 1.00 5 000 2.83E-07 2.00 5.60E-06 1.50 1.93E-05 1.33 3.68E-05 1.25 5.51E-05 1.20 3.23E-04 1.00 8 000 1.10E-07 2.00 2.76E-06 1.50 1.03E-05 1.33 2.05E-05 1.25 3.13E-05 1.20 2.02E-04 1.00

 N L1 L2 L3 L4 L5 L∞ 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 200 2.49E-04 3.48E-04 4.13E-04 4.62E-04 5.02E-04 9.45E-04 300 1.11E-04 1.99 1.55E-04 1.99 1.85E-04 1.99 2.07E-04 1.99 2.24E-04 1.99 4.22E-04 1.99 500 4.01E-05 1.99 5.61E-05 1.99 6.67E-05 1.99 7.47E-05 1.99 8.11E-05 1.99 1.52E-04 2.00 800 1.57E-05 2.00 2.20E-05 2.00 2.61E-05 2.00 2.92E-05 2.00 3.18E-05 2.00 5.94E-05 2.00 1 200 6.98E-06 2.00 9.77E-06 2.00 1.16E-05 2.00 1.30E-05 2.00 1.41E-05 2.00 2.64E-05 2.00 2 000 2.52E-06 2.00 3.52E-06 2.00 4.19E-06 2.00 4.69E-06 2.00 5.09E-06 2.00 9.51E-06 2.00 3 000 1.12E-06 2.00 1.56E-06 2.00 1.86E-06 2.00 2.09E-06 2.00 2.27E-06 2.00 4.23E-06 2.00 5 000 4.03E-07 2.00 5.64E-07 2.00 6.70E-07 2.00 7.51E-07 2.00 8.16E-07 2.00 1.52E-06 2.00 8 000 1.57E-07 2.00 2.20E-07 2.00 2.62E-07 2.00 2.93E-07 2.00 3.19E-07 2.00 5.95E-07 2.00

 $\begin{array}{l} {\rm{if}}\left( {\left| {{x_0}} \right| = 0.4\& {y_0} = 0} \right)\;\;\;\;\;{\rm{then}}\\ \;\;\;\;\;\;{y_{i,j}} = B \cdot {\left( {\Delta x \cdot \Delta y} \right)^{\frac{\gamma }{2}}}\\ {\rm{end}}\;{\rm{if}} \end{array}$ （41）

 $\left\{ {\begin{array}{*{20}{l}} {B = 0.3}&{\gamma = 1}\\ {B = 100}&{\gamma = 2}\\ {B = 20000}&{\gamma = 3} \end{array}} \right.$ （42）

 图 2 局部网格示意图(γ=1) Fig. 2 Local enlarged image of mesh (γ=1)

 $\left\{ {\begin{array}{*{20}{l}} {\gamma = 1}&{{R_1} = - 1}\\ {\gamma = 2}&{{R_1} = 0}\\ {\gamma = 3}&{{R_1} = 1} \end{array}} \right.$ （43）

 $O_m^{\gamma = 1} = \frac{2}{m} - 1$ （44）

 $O_m^{\gamma = 2} = \frac{2}{m}$ （45）

γ=3时，仅有m≥2时，由式(22)和式(23)给出的统计误差的Lm范数的数值精度Omγ=3相同，且均为

 $O_m^{\gamma = 3} = \frac{2}{m} + 1\;\;\;\;\;\left( {m \ge 2} \right)$ （46）

 N L1 L2 L3 L4 L5 L∞ 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 200 1.24E-03 4.98E-02 1.84E-01 3.53E-01 5.22E-01 2.50E+00 300 7.73E-04 1.16 4.98E-02 0 2.10E-01 -0.34 4.32E-01 -0.50 6.66E-01 -0.60 3.75E+00 -1.00 500 4.39E-04 1.11 4.99E-02 0 2.50E-01 -0.34 5.58E-01 -0.50 9.05E-01 -0.60 6.25E+00 -1.00 800 2.65E-04 1.07 4.99E-02 0 2.92E-01 -0.33 7.07E-01 -0.50 1.20E+00 -0.60 1.00E+01 -1.00 1 200 1.73E-04 1.05 5.00E-02 0 3.35E-01 -0.33 8.66E-01 -0.50 1.53E+00 -0.60 1.50E+01 -1.00 2 000 1.02E-04 1.03 5.00E-02 0 3.97E-01 -0.33 1.12E+00 -0.50 2.08E+00 -0.60 2.50E+01 -1.00 3 000 6.77E-05 1.02 5.00E-02 0 4.54E-01 -0.33 1.37E+00 -0.50 2.65E+00 -0.60 3.75E+01 -1.00 5 000 4.04E-05 1.01 5.00E-02 0 5.39E-01 -0.33 1.77E+00 -0.50 3.61E+00 -0.60 6.25E+01 -1.00 8 000 2.52E-05 1.01 5.00E-02 0 6.30E-01 -0.33 2.24E+00 -0.50 4.78E+00 -0.60 1.00E+02 -1.00

 N L1 L2 L3 L4 L5 L∞ 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 200 1.90E-03 8.29E-02 3.06E-01 5.88E-01 8.70E-01 4.17E+00 300 8.47E-04 1.99 5.54E-02 1.00 2.34E-01 0.66 4.80E-01 0.50 7.40E-01 0.40 4.17E+00 0 500 3.06E-04 1.99 3.33E-02 1.00 1.66E-01 0.67 3.72E-01 0.50 6.04E-01 0.40 4.17E+00 0 800 1.20E-04 2.00 2.08E-02 1.00 1.22E-01 0.67 2.94E-01 0.50 5.00E-01 0.40 4.17E+00 0 1 200 5.32E-05 2.00 1.39E-02 1.00 9.29E-02 0.67 2.40E-01 0.50 4.25E-01 0.40 4.17E+00 0 2 000 1.92E-05 2.00 8.33E-03 1.00 6.61E-02 0.67 1.86E-01 0.50 3.47E-01 0.40 4.17E+00 0 3 000 8.52E-06 2.00 5.55E-03 1.00 5.05E-02 0.67 1.52E-01 0.50 2.95E-01 0.40 4.17E+00 0 5 000 3.07E-06 2.00 3.33E-03 1.00 3.59E-02 0.67 1.18E-01 0.50 2.40E-01 0.40 4.17E+00 0 8 000 1.20E-06 2.00 2.08E-03 1.00 2.62E-02 0.67 9.32E-02 0.50 1.99E-01 0.40 4.17E+00 0

 N L1 L2 L3 L4 L5 L∞ 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 200 1.90E-03 8.29E-02 3.06E-01 5.88E-01 8.70E-01 4.17E+00 300 6.02E-04 2.83 3.69E-02 2.00 1.56E-01 1.66 3.20E-01 1.50 4.93E-01 1.40 2.78E+00 1.00 500 1.46E-04 2.77 1.33E-02 2.00 6.66E-02 1.66 1.49E-01 1.50 2.41E-01 1.40 1.67E+00 1.00 800 4.17E-05 2.67 5.20E-03 2.00 3.04E-02 1.67 7.36E-02 1.50 1.25E-01 1.40 1.04E+00 1.00 1 200 1.47E-05 2.57 2.31E-03 2.00 1.55E-02 1.67 4.01E-02 1.50 7.09E-02 1.40 6.94E-01 1.00 2 000 4.18E-06 2.46 8.33E-04 2.00 6.61E-03 1.67 1.86E-02 1.50 3.47E-02 1.40 4.17E-01 1.00 3 000 1.61E-06 2.35 3.70E-04 2.00 3.36E-03 1.67 1.01E-02 1.50 1.97E-02 1.40 2.78E-01 1.00 5 000 5.09E-07 2.26 1.33E-04 2.00 1.44E-03 1.67 4.71E-03 1.50 9.62E-03 1.40 1.67E-01 1.00 8 000 1.83E-07 2.17 5.21E-05 2.00 6.56E-04 1.67 2.33E-03 1.50 4.98E-03 1.40 1.04E-01 1.00
2.4 算例应用

 $u\left( x \right) = \sin\left[ {2{\rm{ \mathsf{ π} }}x - \frac{{\sin\left( {2{\rm{ \mathsf{ π} }}x} \right)}}{{2{\rm{ \mathsf{ π} }}}}} \right]$ （47）

 ${O_m} = 3 + \frac{1}{m}$ （48）

 N L1 L2 L3 L4 L5 L∞ 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 误差 精度 200 8.88E-08 3.01E-07 5.87E-07 8.46E-07 1.06E-06 2.67E-06 300 1.64E-08 4.17 7.66E-08 3.38 1.67E-07 3.10 2.52E-07 2.99 3.24E-07 2.93 8.84E-07 2.73 500 1.90E-09 4.22 1.26E-08 3.53 3.04E-08 3.33 4.80E-08 3.25 6.33E-08 3.20 1.91E-07 3.00 800 2.98E-10 3.94 2.12E-09 3.80 4.88E-09 3.89 7.57E-09 3.93 9.90E-09 3.95 3.07E-08 3.89 1 200 5.77E-11 4.05 5.11E-10 3.51 1.26E-09 3.35 2.01E-09 3.28 2.67E-09 3.23 8.61E-09 3.13 2 000 7.36E-12 4.03 8.54E-11 3.50 2.29E-10 3.33 3.81E-10 3.25 5.20E-10 3.20 1.85E-09 3.01 3 000 1.44E-12 4.02 2.07E-11 3.50 5.95E-11 3.32 1.03E-10 3.23 1.44E-10 3.17 5.78E-10 2.87 5 000 1.85E-13 4.02 3.49E-12 3.49 1.10E-11 3.30 2.01E-11 3.19 2.91E-11 3.13 1.35E-10 2.85 8 000 2.80E-14 4.02 6.85E-13 3.46 2.40E-12 3.25 4.64E-12 3.13 6.94E-12 3.05 3.61E-11 2.80

3 结论

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http://dx.doi.org/10.7527/S1000-6893.2019.23554

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#### 文章信息

MIN Yaobing, MA Yankai, LI Song
CFD中统计误差的数值精度分析
Accuracy analysis of numerical error with statistical forms in CFD

Acta Aeronautica et Astronautica Sinica, 2020, 41(4): 123554.
http://dx.doi.org/10.7527/S1000-6893.2019.23554