Some computational aspects of approximate inertial manifolds and finite differences

An approach to the concept of approximate inertial manifolds for dissipative evolutionary equations in combination with finite difference semidiscretizations is presented. We introduce general frequency decompositions of the underlying finite dimensional solution space and consider the inertial form corresponding to this decomposition. It turns out that, under certain restrictions, all terms in the inertial form can be explicitly expanded as functions of the new coefficients. The calculations are carried out for reaction diffusion equations in 1D, 2D and 3D and for the Kuramoto-Sivashinsky equation in 1D, and numerical results are presented.