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

Antonios Zagaris, C.W. Gear, T.J. Kaper, I.G. Kevrekidis

25 Citations (Scopus)

Abstract

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the mth member of the class of algorithms ( $m = 0, 1, \ldots$) finds iteratively an approximation of the appropriate zero of the (m+1)st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with fast-slow systems, in which there is an explicit small parameter, $\varepsilon$, measuring the separation of time scales. We show that, for each $m = 0, 1, \ldots$, the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of ${\mathcal O}(\varepsilon^m)$. Moreover, for each m, we identify explicitly the conditions under which the mth iterative algorithm converges to this fixed point. Finally, we show that when the iteration is unstable (or converges slowly) it may be stabilized (or its convergence may be accelerated) by application of the Recursive Projection Method. Alternatively, the Newton-Krylov Generalized Minimal Residual Method may be used. In the subsequent article, we will consider the accuracy and convergence of the iterative algorithms for a broader class of systems – in which there need not be an explicit small parameter – to which the algorithms also apply.
Original language Undefined 10.1051/m2an/2009026 757-784 28 ESAIM: Mathematical Modelling and Numerical Analysis 43 4 https://doi.org/10.1051/m2an/2009026 Published - Jul 2009

Keywords

• MSC-65P99
• MSC-65L20
• MSC-35B25
• IR-69756
• MSC-37M99
• EWI-17261
• METIS-265769
• MSC-35B42

Cite this

Zagaris, Antonios ; Gear, C.W. ; Kaper, T.J. ; Kevrekidis, I.G. / Analysis of the accuracy and convergence of equation-free projection to a slow manifold. In: ESAIM: Mathematical Modelling and Numerical Analysis. 2009 ; Vol. 43, No. 4. pp. 757-784.
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Analysis of the accuracy and convergence of equation-free projection to a slow manifold. / Zagaris, Antonios; Gear, C.W.; Kaper, T.J.; Kevrekidis, I.G.

In: ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 43, No. 4, 10.1051/m2an/2009026, 07.2009, p. 757-784.

TY - JOUR

T1 - Analysis of the accuracy and convergence of equation-free projection to a slow manifold

AU - Zagaris, Antonios

AU - Gear, C.W.

AU - Kaper, T.J.

AU - Kevrekidis, I.G.

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AB - In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the mth member of the class of algorithms ( $m = 0, 1, \ldots$) finds iteratively an approximation of the appropriate zero of the (m+1)st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with fast-slow systems, in which there is an explicit small parameter, $\varepsilon$, measuring the separation of time scales. We show that, for each $m = 0, 1, \ldots$, the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of ${\mathcal O}(\varepsilon^m)$. Moreover, for each m, we identify explicitly the conditions under which the mth iterative algorithm converges to this fixed point. Finally, we show that when the iteration is unstable (or converges slowly) it may be stabilized (or its convergence may be accelerated) by application of the Recursive Projection Method. Alternatively, the Newton-Krylov Generalized Minimal Residual Method may be used. In the subsequent article, we will consider the accuracy and convergence of the iterative algorithms for a broader class of systems – in which there need not be an explicit small parameter – to which the algorithms also apply.

KW - MSC-65P99

KW - MSC-65L20

KW - MSC-35B25

KW - IR-69756

KW - MSC-37M99

KW - EWI-17261

KW - METIS-265769

KW - MSC-35B42

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DO - 10.1051/m2an/2009026

M3 - Article

VL - 43

SP - 757

EP - 784

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

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M1 - 10.1051/m2an/2009026

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