### 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.

Original language | English (US) |
---|---|

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 |

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### ASJC Scopus subject areas

- Computer Science Applications
- Signal Processing
- Analysis
- Control and Optimization

### Cite this

**On the Behavior of Attractors Under Finite Difference Approximation.** / Jones, Donald.

Research output: Contribution to journal › Article

*Numerical Functional Analysis and Optimization*, vol. 16, no. 9-10, pp. 1155-1180. https://doi.org/10.1080/01630569508816667

}

TY - JOUR

T1 - On the Behavior of Attractors Under Finite Difference Approximation

AU - Jones, Donald

PY - 1995/1/1

Y1 - 1995/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0011601917&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011601917&partnerID=8YFLogxK

U2 - 10.1080/01630569508816667

DO - 10.1080/01630569508816667

M3 - Article

AN - SCOPUS:0011601917

VL - 16

SP - 1155

EP - 1180

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 9-10

ER -