COMPUTATIONAL-EFFICIENCY AND APPROXIMATE INERTIAL MANIFOLDS FOR A BENARD CONVECTION SYSTEM

A computational comparison between classical Galerkin and approximate inertial manifold (AIM) methods is performed for the case of two-dimen sional natural convection in a saturated porous material. For predicti on of Hopf and torus bifurcations far from convection onset, the impro vements of the AIM method over the classical one are small or negligib le. Two reasons are given for the lack of distinct improvement. First, the small boundary layer length scale is the source of the instabilit ies, so it cannot be modeled as a ''slave'' to the larger scales, as t he AIM attempts to do. Second, estimates based on the Gevrey class reg ularity of solutions to the governing equations show that the classica l and AIM methods may be virtually equivalent. It is argued that these two reasons are physical and mathematical reflections of one another.