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Model Reduction of the Nonlinear Complex Ginzburg-Landau Equation
KTH, School of Engineering Sciences (SCI), Mechanics, Stability, Transition and Control.
KTH, School of Engineering Sciences (SCI), Mechanics, Stability, Transition and Control.ORCID iD: 0000-0002-8209-1449
KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0002-4346-4732
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2010 (English)In: SIAM Journal on Applied Dynamical Systems, ISSN 1536-0040, E-ISSN 1536-0040, Vol. 9, no 4, 1284-1302 p.Article in journal (Refereed) Published
Abstract [en]

Reduced-order models of the nonlinear complex Ginzburg-Landau (CGL) equation are computed using a nonlinear generalization of balanced truncation. The method involves Galerkin projection of the nonlinear dynamics onto modes determined by balanced truncation of a linearized system and is compared to a standard method using projection onto proper orthogonal decomposition (POD) modes computed from snapshots of nonlinear simulations. It is found that the nonlinear reduced-order models obtained using modes from linear balanced truncation capture very well the transient dynamics of the CGL equation and outperform POD models; i.e., a higher number of POD modes than linear balancing modes is typically necessary in order to capture the dynamics of the original system correctly. In addition, we find that the performance of POD models compares well to that of balanced truncation models when the degree of nonnormality in the system, in this case determined by the streamwise extent of a disturbance amplification region, is lower. Our findings therefore indicate that the superior performance of balanced truncation compared to POD/Galerkin models in capturing the input/output dynamics of linear systems extends to the case of a nonlinear system, both for the case of significant transient growth, which represents a basic model of boundary layer instabilities, and for a limit cycle case that represents a basic model of vortex shedding past a cylinder.

Place, publisher, year, edition, pages
2010. Vol. 9, no 4, 1284-1302 p.
Keyword [en]
model reduction, balanced truncation, Ginzburg-Landau equation
National Category
Computational Mathematics Other Physics Topics
URN: urn:nbn:se:kth:diva-29354DOI: 10.1137/100787350ISI: 000285550500007ScopusID: 2-s2.0-79251556852OAI: diva2:395435
Swedish Research CouncilSwedish e‐Science Research Center
QC 20110207Available from: 2011-02-07 Created: 2011-02-01 Last updated: 2012-05-22Bibliographically approved

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