The Problem With Oscillatory Behavior in Grid Convergence Studies

Abstract

Contributed by the Fluids Engineering Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Fluids Engineering Division September 1, 1999; revised manuscript received February 8, 2001. Associate Editor: V. Ghia. The possibility of oscillatory behavior of the value of a computed variable as grid size is refined in a simulation raises questions in the interpretation of grid convergence studies that have, to the authors’ knowledge, never been addressed. Such oscillatory behavior has been observed by the authors, for example, in variables such as wave profiles along the hull and wave elevations in the stern flow in simulations of flows about ships with complex geometries. Roache 1, in his comprehensive presentation and critique of work up to that time in the area of verification and validation of simulations, points out that “behavior far away from asymptotic convergence can be non-monotone” and “the additional assumption of monotone truncation error convergence in the mesh spacing…may not be valid for coarse grids, or possibly other conditions.” Typically (see Stern, et al. 2, for instance), the behavior of a variable is categorized as monotonically convergent, oscillatory, or divergent based on its behavior as the grid used in a simulation is refined. Consider the values of a computed variable y as a simulation is run using a coarse grid yC, a medium grid yM, and a fine grid yF. The ratio R=yM−yFyC−yM=ΔyM−FΔyC−Mhas been used to categorize the behavior of y as grid size is decreased as: (1) monotonically convergent when 0<R<1; (2) oscillatory when R<0; and (3) divergent when R>1. If the behavior really is monotonically convergent, then (1) holds. If the behavior really is divergent, then (3) holds. The problem that has not been previously recognized and discussed is the ambiguity that arises when the behavior really is oscillatory. Consider the results from a simulation in which the computed value of variable y is truly an oscillatory function of grid size Δx, as shown in Fig. 1. When a grid size is chosen in a simulation, its value is of course arbitrary relative to the unknown periods of any oscillations of the computed variables, so each of the following cases in this example must be considered equally likely. Three cases are investigated, with a different initial grid size in each. In each case, three simulations are run with grid doubling used twice, resulting in coarse (C), medium (M), and fine (F) grid simulation values of the variable y. The computed values of y are shown in Table 1 and plotted in Fig. 1. For Case 1, ΔyC−M is −1.0 and ΔyM−F is +0.3, a situation that would be assessed as oscillatory since R<0. For Case 2, ΔyC−M is +0.36 and ΔyM−F is +0.28, a situation that would be concluded as being monotonically convergent since 0<R<1. Finally, for Case 3, ΔyC−M is +0.14 and ΔyM−F is +0.36, a situation that would be concluded as divergent since R>1. Thus, for the same (true) oscillatory behavior any of three conclusions can be supported, depending on the relationship of the chosen grid size to the unknown period(s) of the oscillation(s). Note that in real cases (with results from three grids, say), the true behavior of y with Δx is unknown, so the only information one has is the three computed values of y. If the true but unknown behavior is oscillatory, then depending on the choices of initial grid size and the grid refinement ratio those three values can produce any value of R (including a value indicating asymptotic convergence). Also note that whether the Δx in one’s finest grid corresponds to a value of 1, 2, 3, or 100 on a scale such as that shown in Fig. 1 is unknown. Although the example presented is somewhat contrived, the dilemma one faces in interpreting results of grid convergence studies is not. If there is the possibility of oscillatory behavior of the value of a computed variable as grid size is refined in a simulation, then interpretation of the results of grid convergence studies seems impossible to achieve unambiguously. The authors gratefully acknowledge the sponsorship of this research by the Office of Naval Research under Grants N00014-97-1-0014 and N00014-97-1-0151 under the administration of Dr. E. P. Rood.