On the behavior of attractors under finite difference approximation |
| |
Authors: | Don A. Jones |
| |
Affiliation: | IGPP Univeristy of California Los Alamos National Laboratory , Mail Stop C305, Los Alamos, NM, 87544 |
| |
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 C 1 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. |
| |
Keywords: | 34C40 34D45 58F30 58F32 35K55 65M06 |
|
|