Equation-free, effective computation for discrete systems: A time stepper based approach

J. Möller; O. Runborg; P. G. Kevrekidis; K. Lust; I. G. Kevrekidis

doi:10.1142/S0218127405012399

Equation-free, effective computation for discrete systems: A time stepper based approach

J. Möller, O. Runborg, P. G. Kevrekidis, K. Lust, I. G. Kevrekidis

Chemical & Biological Engineering

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Padé approximations.

Original language	English (US)
Pages (from-to)	975-996
Number of pages	22
Journal	International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume	15
Issue number	3
DOIs	https://doi.org/10.1142/S0218127405012399
State	Published - Mar 2005

All Science Journal Classification (ASJC) codes

Modeling and Simulation
Engineering (miscellaneous)
General
Applied Mathematics

Keywords

Bifurcation
Discrete problems
Equation-free methods
Homogenization
Pinning condition

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10.1142/S0218127405012399

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title = "Equation-free, effective computation for discrete systems: A time stepper based approach",

abstract = "We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the {"}coarse model{"} (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Pad{\'e} approximations.",

keywords = "Bifurcation, Discrete problems, Equation-free methods, Homogenization, Pinning condition",

author = "J. M{\"o}ller and O. Runborg and Kevrekidis, {P. G.} and K. Lust and Kevrekidis, {I. G.}",

note = "Funding Information: Part of the research for this paper was carried out while Olof Runborg held a post-doctoral appointment with the Program for Applied and Computational Mathematics at Princeton University, supported by NSF KDI grant DMS-9872890. Panayotis G. Kevrekidis gratefully acknowledges support from a UMass FRG, NSF-DMS-0204585 and from the Eppley Foundation for Research. Kurt Lust is a postdoctoral fellow of the Fund for Scientific Research–Flanders. This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister{\textquoteright}s Office for Science, Technology and Culture. The scientific Funding Information: responsibility rests with its authors. Ioannis G. Kevrekidis gratefully acknowledges the support of AFOSR (Dynamics and Control) and an NSF-ITR grant.",

year = "2005",

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doi = "10.1142/S0218127405012399",

language = "English (US)",

volume = "15",

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AU - Runborg, O.

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AU - Lust, K.

AU - Kevrekidis, I. G.

N1 - Funding Information: Part of the research for this paper was carried out while Olof Runborg held a post-doctoral appointment with the Program for Applied and Computational Mathematics at Princeton University, supported by NSF KDI grant DMS-9872890. Panayotis G. Kevrekidis gratefully acknowledges support from a UMass FRG, NSF-DMS-0204585 and from the Eppley Foundation for Research. Kurt Lust is a postdoctoral fellow of the Fund for Scientific Research–Flanders. This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture. The scientific Funding Information: responsibility rests with its authors. Ioannis G. Kevrekidis gratefully acknowledges the support of AFOSR (Dynamics and Control) and an NSF-ITR grant.

PY - 2005/3

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N2 - We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Padé approximations.

AB - We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Padé approximations.

KW - Bifurcation

KW - Discrete problems

KW - Equation-free methods

KW - Homogenization

KW - Pinning condition

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Equation-free, effective computation for discrete systems: A time stepper based approach

Abstract

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