Some computational aspects of approximate inertial manifolds and finite differences. (English) Zbl 0949.65135

Summary: 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.


65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37M05 Simulation of dynamical systems
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35K57 Reaction-diffusion equations
35Q58 Other completely integrable PDE (MSC2000)
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