Abstract
We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C1 perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.
Original language | English (US) |
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Pages (from-to) | 1155-1180 |
Number of pages | 26 |
Journal | Numerical Functional Analysis and Optimization |
Volume | 16 |
Issue number | 9-10 |
DOIs | |
State | Published - Jan 1 1995 |
Externally published | Yes |
ASJC Scopus subject areas
- Analysis
- Signal Processing
- Computer Science Applications
- Control and Optimization