On the effectiveness of the approximate inertial manifold?a computational study

Abstract

We present a computational study evaluating the effectiveness of the nonlinear Galerkin method for dissipative evolution equations. We begin by reviewing the theoretical estimates of the rate of convergence for both the standard spectral Galerkin and the nonlinear Galerkin methods. We show that the rate of convergence in both cases depends mainly on how well the basis functions of the spectral method approximate the elements in the space of solutions. This in turn depends on the degree of smoothness of the basis functions, the smoothness of the solutions, and on the level of compatibility at the boundary between the basis functions of the spectral method and the solutions. When the solutions are very smooth inside the domain and very compatible with the basis functions at the boundary, there may be little advantage in using the nonlinear Galerkin method. On the other hand, for less smooth solutions or when there is less compatibility at the boundary with the basis functions, there is a significant improvement in the rate of convergence when using the nonlinear Galerkin method. We demonstrate the validity of our assertions with numerical simulations of the forced dissipative Burgers equation and of the forced Kuramoto-Sivashinsky equation. These simulations also demonstrate that the analytical upper bounds derived for the rates of convergence of both the standard Galerkin and the nonlinear Galerkin are nearly sharp.