Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Some computational aspects of approximate inertial manifolds and finite differences

Pages: 417 - 454, Volume 2, Issue 4, October 1996      doi:10.3934/dcds.1996.2.417

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Rolf Bronstering - Institut für Numerische Mathematik, Universität Münster, Germany (email)

Abstract: 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.

Keywords:  dissipative evolutionary equations, reaction diffusion, Kuramoto-Sivashinsky equation.
Mathematics Subject Classification:  35A40, 65M06.

Received: August 1995;      Revised: May 1996;      Available Online: July 1996.