Analysis of commutation errors for functions with low regularity

Commutation errors arise in the derivation of the space averaged Navier–Stokes equations, the basic equations for the large eddy simulation of turbulent flows, if the filter is non-uniform or asymmetric (skewed) with non-constant skewness. These errors need to be analyzed for turbulent flow fields, where one expects a limited regularity of the solution. This paper studies the order of convergence of commutation errors, as the filter width tends to zero, for functions with low regularity. Several convergence results are proved and it is also shown that convergence may fail (or its order decreases) if the functions become less smooth. The main results are those dealing with Hölder–continuous functions and with functions having singularities. The sharpness of the analytic results is confirmed with numerical illustrations.