Evaluation of Extrapolation-Based Discretization Error and Uncertainty Estimators

Abstract

The largest and most difficult numerical approximation error to estimate is discretization error. This study investigates the accuracy of various Richardson extrapolation-based discretization error and uncertainty estimators for problems in computational fluid dynamics. Richardson extrapolation uses two solutions on systematically refined grids to estimate the exact solution to the partial differential equations and is accurate only in the asymptotic range (i.e., when the grids are sufficiently fine). The uncertainty estimators that will be investigated are different implementations and the Grid Convergence Index and include a globally averaged observed order of accuracy, a least squares calculation of the observed order of accuracy, the Factor of Safety method, and the Correction Factor method. Two twodimensional, inviscid, supersonic flow fields with exact solutions and a twodimensional, turbulent flat plate are used to evaluate the extrapolation-based discretization error and uncertainty estimators. The conservativeness (percent of cases where the exact solution is bracketed by the estimated uncertainties) and the effectivity (how accurately the discretization error estimate approximates the exact error) are used to assess the relative merits of the different approaches. The overall trend suggests a trade-off between conservativeness and effectivity (e.g. the most conservative uncertainty estimator was the least accurate and vise-versa). The least squares method showed the best trade-off between conservativeness and effecitivity under specific conditions.