Efficient methods using high accuracy approximate inertial manifolds

We extend the idea of the post-processing Galerkin method, in the context o f 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 land 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 ch osen, highly accurate, approximate inertial manifold (AIM). We present nume rical experiments that support the theoretical improvements in accuracy. Bo th the theory and computations are presented in the framework of a two dime nsional reaction-diffusion system with polynomial nonlinearity. However, th e algorithm is very general and can be implemented for other dissipative ev olution systems. The computations clearly show the post-processed filtered Galerkin method to be the most efficient method.