Summary
A computational comparison between classical Galerkin and approximate inertial manifold (AIM) methods is performed for the case of two-dimensional natural convection in a saturated porous material. For prediction of Hopf and torus bifurcations far from convection onset, the improvements of the AIM method over the classical one are small or negligible. Two reasons are given for the lack of distinct improvement. First, the small boundary layer length scale is the source of the instabilities, so it cannot be modeled as a “slave” to the larger scales, as the AIM attempts to do. Second, estimates based on the Gevrey class regularity of solutions to the governing equations show that the classical and AIM methods may be virtually equivalent. It is argued that these two reasons are physical and mathematical reflections of one another.
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Communicated by Ioannis Kevrekidis
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Graham, M.D., Steen, P.H. & Titi, E.S. Computational efficiency and approximate inertial manifolds for a Bénard convection system. J Nonlinear Sci 3, 153–167 (1993). https://doi.org/10.1007/BF02429862
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DOI: https://doi.org/10.1007/BF02429862