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We revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view. We assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation. Keeping terms to certain orders, we then generate approximate systems which correspond to numerical schemes. We show that the first two leading order methods are in fact the postprocessed Galerkin and postprocessed nonlinear Galerkin methods, respectively. Hence postprocessed Galerkin is a natural leading order method, more natural than the standard Galerkin method, for approximating solutions of parabolic dissipative PDEs. The analysis is presented in the framework of the two-dimensional Navier--Stokes equation (NSE); however, similar analysis may be done for any parabolic, dissipative nonlinear PDE.The truncation analysis is based on asymptotic estimates (in time) for the low and high mode components. We also introduce and investigate an alternative postprocessing scheme, whic...

In this paper we will survey our results on the Camassa–Holm equations and their relation to turbulence as discussed in S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, The Camassa–Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett 81 (1998) 5338. S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, A connection between the Camassa–Holm equations and turbulent flows in channels and pipes, Phys. Fluids, in press. In particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large Reynolds numbers. There are many facts connecting the Camassa–Holm equations to turbulent fluid flows. The dimension of the attractor agrees with the heuristic argument based on the Kolmogorov statistical theory of turbulence. The statistical properties of the energy spectrum agree in numerical simulation with the Kolmogorov power law. Furthermore, comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the Camassa–Holm equations.

Summary. We extend the idea of the post-processing Galerkin method, in the context of dissipative evolution equations, to the nonlinear Galerkin, the filtered Galerkin, and the filtered nonlinear Galerkin methods. In general, the post-processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear Galerkin method (and Galerkin method), for approximating smooth, bounded solutions, is much smaller than the total error of the method. In each case, an improvement in accuracy is obtained by post-processing these more accurate lower modes with an appropriately chosen, highly accurate, approximate inertial manifold (AIM). We present numerical experiments that support the theoretical improvements in accuracy. Both the theory and computations are presented in the framework of a two dimensional reaction-diffusion system with polynomial nonlinearity. However, the algorithm is very general and can be implemented for other dissipative evolution systems. The computations clearly show the post-processed filtered Galerkin method to be the most efficient method.