On the Influence of the Iterative Error in the Numerical Uncertainty of Ship Viscous Flow Calculations

This paper presents a study on the influence of the iterative error on the numerical uncertainty of the solution of the Reynolds-averaged Navier-Stokes equations. Two main topics are addressed: the estimation of the iterative error; and the influence of the iterative error on the estimation of the discretization error.
Iterative error estimators based on the L∞, L ₁ and L ₂ norms of the differences between iterations and on the normalized residuals are tested on three test cases: the 2-D turbulent flow over a hill, a 3-D flow over a finite plate and the flow around the KVLCC2M tanker at model scale Reynolds number. Two types of procedures are considered, one using the data of the last iteration performed and the other using an extrapolation based on a least squares fit to a geometric progression. In the latter case, the option of including the standard deviation of the fit in calculation of the error estimator is also tested.
The results show that the most reliable estimates of the iterative error are obtained with the extrapolation technique including the effect of the standard deviation of the fit applied to the L∞ norm of the differences between successive solutions. To obtain realistic estimates of the iterative error one should avoid the use of the L ₂ and L ₁ norms and the use of the differences obtained in the last iteration.
The results further suggest that the contributions of the iterative and discretization errors to the numerical uncertainty are not independent. Nevertheless, for the levels of grid refinement tested it is not necessary to converge a solution to machine accuracy to obtain a negligible contribution of the iterative error. It suffices to ensure that the convergence criterion guarantees an iterative error roughly 2 to 3 orders of magnitude below the discretization error.