On a Comparison of Solutions in Verification Problems

摘要

— 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.