Abstract—

The impact of the choice of the proximity measure for the numerical and reference solutions is discussed in terms of the verification of the calculations and software. If no reference solution is available, the deterministic and stochastic options for estimating computational errors are considered using an ensemble of solutions obtained by different numerical algorithms. The relation between the norm of the solution error and the error of valuable functionals is studied via the Cauchy–Bunyakovsky–Schwarz inequality. The results of numerical tests for the two-dimensional Euler equations, which demonstrate how the choice of the proximity measure affects the estimation of the approximation error on the ensemble of solutions and show the efficiency of the considered algorithms, are presented. The comparison of different proximity measures (norms and metrics) both for estimating the computational error and for comparing the flow fields that correspond to both small variations in the flow structure and qualitatively different flow patterns is a new element of the paper. The application of the errors of valuable functionals for the evaluation of the approximation errors in practical terms is also novel. The feasibility for computationally cheap (single-grid, in contrast to the Richardson extrapolation method) quantitative verification of solutions considered and analyzed in the paper seems useful for the implementation of the Russian standards for numerical solution verification and CFD code validation.

中文翻译：

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验证问题解决方案的比较

摘要—

根据计算和软件的验证，讨论了选择邻近度量对数值和参考解的影响。如果没有可用的参考解，则使用由不同数值算法获得的解的整体来考虑用于估计计算误差的确定性和随机性选项。通过Cauchy–Bunyakovsky–Schwarz不等式研究了解误差范数与有价值函数的误差之间的关系。给出了二维Euler方程的数值测试结果，这些结果证明了邻近度量的选择如何影响解决方案集合中的近似误差估计，并表明了所考虑算法的效率。为了估计计算误差和比较与流动结构的小变化和定性不同的流动模式相对应的流场，对不同接近度度量（标准和度量）进行比较是本文的一个新内容。在实用上将有价值的函数的误差用于近似误差的评估也是新颖的。本文中考虑和分析的解决方案的计算廉价（单网格，与Richardson外推法相反）的定量验证的可行性似乎对实施俄罗斯数字解决方案验证和CFD代码验证标准很有用。