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Analysis of the accuracy and convergence of equation-free projection to a slow manifold

Published online by Cambridge University Press:  08 July 2009

Antonios Zagaris ,
C. William Gear ,
Tasso J. Kaper  and
Yannis G. Kevrekidis
Antonios Zagaris
Affiliation:
Department of Mathematics, University of Amsterdam, Amsterdam, The Netherlands. Modeling, Analysis and Simulation, Centrum Wiskunde & Informatica, Amsterdam, The Netherlands.
C. William Gear
Affiliation:
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA. NEC Laboratories USA, retired.
Tasso J. Kaper
Affiliation:
Department of Mathematics and Center for BioDynamics, Boston University, Boston, MA 02215, USA. [email protected]
Yannis G. Kevrekidis
Affiliation:
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA. Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA.
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Abstract

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris,SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732],we developeda class of iterative algorithmswithin the contextof equation-free methodsto approximatelow-dimensional,attracting,slow manifoldsin systemsof differential equationswith multiple time scales.For user-specified valuesof a finite numberof the observables,the mth memberof the classof algorithms( $m = 0, 1, \ldots$ )finds iterativelyan approximationof the appropriate zeroof the (m+1)st time derivativeof the remaining variablesanduses this rootto approximate the locationof the pointon the slow manifoldcorresponding to these valuesof the observables.This articleis the firstof two articlesin whichthe accuracy and convergenceof the iterative algorithmsare analyzed.Here,we work directlywith fast-slow systems,in which there isan explicit small parameter, ε,measuring the separationof time scales.We show that,for each $m = 0, 1, \ldots$ ,the fixed pointof the iterative algorithmapproximates the slow manifoldup to and includingterms of ${\mathcal O}(\varepsilon^m)$ .Moreover,for each m,we identify explicitlythe conditionsunder whichthe mth iterative algorithmconverges to this fixed point.Finally,we show thatwhenthe iterationis unstable(orconverges slowly)it may be stabilized(orits convergencemay be accelerated)by applicationof the Recursive Projection Method.Alternatively,the Newton-KrylovGeneralized Minimal Residual Methodmay be used.In the subsequent article,we will considerthe accuracy and convergenceof the iterative algorithmsfor a broader classof systems – in whichthere need not bean explicitsmall parameter – to whichthe algorithms also apply.

Keywords

Iterative initializationDAEssingular perturbationslegacy codesinertial manifolds.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 4: Special issue on Numerical ODEs today , July 2009 , pp. 757 - 784
Copyright
© EDP Sciences, SMAI, 2009

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A. Zagaris, C. Vandekerckhove, C.W. Gear, T.J. Kaper and I.G. Kevrekidis, Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold. Numer. Math. (submitted).