Additive turbulent decomposition for the two-dimensional incompressible Navier-Stokes equations: convergence theorems and error estimates

The additive turbulent decomposition (ATD) method is a computational scheme for solving the Navier-Stokes equations and related nonlinear dissipative evolution equations. It involves a decomposition of the Navier-Stokes equation into equations for large- and small-scale components similar in spirit, but different in details, to the nonlinear Galerkin methods proposed by Temam and coworkers. In this paper, we consider the ATD method as applied to the two-dimensional incompressible Navier-Stokes equation on a bounded domain. We model the corresponding ATD equations by a coupled system of nonlinear differential equations, and obtain convergence and error estimates. The error estimates for the ATD method are comparable to those for the linear Galerkin method. We show how to modify the ATD method to obtain a scheme for which the convergence estimates are similar to those of the first nonlinear Galerkin method.