Numerical computation of ε-entropy for parabolic equations with analytic solutions

G. J. Lord, J. Rougemont

Research output: Contribution to journalArticle

Abstract

We report on numerical experiments computing the e-entropy for parabolic partial differential equations. The e-entropy is a measure of the spatial density of complexity for the dynamics on an invariant set in function space and has been studied analytically by a number of authors. The e-entropy only requires solutions of the equation to the accuracy of the parameter e and the resulting number is (asymptotically) independent of domain size. We consider the complex Ginzburg-Landau equation as an example where a number of analytic results exist and the Kuramoto-Sivashinsky equation where the accompanying theory has yet to be fully developed. Our numerical results for the Kuramoto-Sivashinsky equation do not contradict the conjectured linear scaling of the dimension with domain size. © 2004 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)65-74
Number of pages10
JournalPhysica D: Nonlinear Phenomena
Volume194
Issue number1-2
DOIs
Publication statusPublished - 1 Jul 2004

Fingerprint

Analytic Solution
Numerical Computation
Parabolic Equation
Kuramoto-Sivashinsky Equation
Entropy
Complex Ginzburg-Landau Equation
Parabolic Partial Differential Equations
Invariant Set
Function Space
Numerical Experiment
Scaling
Numerical Results
Computing

Keywords

  • ε-Entropy
  • Attractor
  • Dimension
  • Gevrey regularity

Cite this

@article{c2e2671d011c421b876ea16bb1e0c785,
title = "Numerical computation of ε-entropy for parabolic equations with analytic solutions",
abstract = "We report on numerical experiments computing the e-entropy for parabolic partial differential equations. The e-entropy is a measure of the spatial density of complexity for the dynamics on an invariant set in function space and has been studied analytically by a number of authors. The e-entropy only requires solutions of the equation to the accuracy of the parameter e and the resulting number is (asymptotically) independent of domain size. We consider the complex Ginzburg-Landau equation as an example where a number of analytic results exist and the Kuramoto-Sivashinsky equation where the accompanying theory has yet to be fully developed. Our numerical results for the Kuramoto-Sivashinsky equation do not contradict the conjectured linear scaling of the dimension with domain size. {\circledC} 2004 Elsevier B.V. All rights reserved.",
keywords = "ε-Entropy, Attractor, Dimension, Gevrey regularity",
author = "Lord, {G. J.} and J. Rougemont",
year = "2004",
month = "7",
day = "1",
doi = "10.1016/j.physd.2004.01.040",
language = "English",
volume = "194",
pages = "65--74",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "1-2",

}

Numerical computation of ε-entropy for parabolic equations with analytic solutions. / Lord, G. J.; Rougemont, J.

In: Physica D: Nonlinear Phenomena, Vol. 194, No. 1-2, 01.07.2004, p. 65-74.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Numerical computation of ε-entropy for parabolic equations with analytic solutions

AU - Lord, G. J.

AU - Rougemont, J.

PY - 2004/7/1

Y1 - 2004/7/1

N2 - We report on numerical experiments computing the e-entropy for parabolic partial differential equations. The e-entropy is a measure of the spatial density of complexity for the dynamics on an invariant set in function space and has been studied analytically by a number of authors. The e-entropy only requires solutions of the equation to the accuracy of the parameter e and the resulting number is (asymptotically) independent of domain size. We consider the complex Ginzburg-Landau equation as an example where a number of analytic results exist and the Kuramoto-Sivashinsky equation where the accompanying theory has yet to be fully developed. Our numerical results for the Kuramoto-Sivashinsky equation do not contradict the conjectured linear scaling of the dimension with domain size. © 2004 Elsevier B.V. All rights reserved.

AB - We report on numerical experiments computing the e-entropy for parabolic partial differential equations. The e-entropy is a measure of the spatial density of complexity for the dynamics on an invariant set in function space and has been studied analytically by a number of authors. The e-entropy only requires solutions of the equation to the accuracy of the parameter e and the resulting number is (asymptotically) independent of domain size. We consider the complex Ginzburg-Landau equation as an example where a number of analytic results exist and the Kuramoto-Sivashinsky equation where the accompanying theory has yet to be fully developed. Our numerical results for the Kuramoto-Sivashinsky equation do not contradict the conjectured linear scaling of the dimension with domain size. © 2004 Elsevier B.V. All rights reserved.

KW - ε-Entropy

KW - Attractor

KW - Dimension

KW - Gevrey regularity

UR - http://www.scopus.com/inward/record.url?scp=2942555578&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2004.01.040

DO - 10.1016/j.physd.2004.01.040

M3 - Article

VL - 194

SP - 65

EP - 74

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -