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

Antonios Zagaris; C. William Gear; Tasso J. Kaper; Yannis G. Kevrekidis

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 4, page 757-784
- ISSN: 0764-583X

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topZagaris, Antonios, et al. "Analysis of the accuracy and convergence of equation-free projection to a slow manifold." ESAIM: Mathematical Modelling and Numerical Analysis 43.4 (2009): 757-784. <http://eudml.org/doc/250588>.

@article{Zagaris2009,

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, ε,
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.
},

author = {Zagaris, Antonios, Gear, C. William, Kaper, Tasso J., Kevrekidis, Yannis G.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Iterative initialization;
DAEs;
singular perturbations;
legacy codes;
inertial manifolds.; iterative initialization; daes; legacy codes; recursive projection method; Newton-Krylov generalized minimal residual method},

language = {eng},

month = {7},

number = {4},

pages = {757-784},

publisher = {EDP Sciences},

title = {Analysis of the accuracy and convergence of equation-free projection to a slow manifold},

url = {http://eudml.org/doc/250588},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Zagaris, Antonios

AU - Gear, C. William

AU - Kaper, Tasso J.

AU - Kevrekidis, Yannis G.

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2009/7//

PB - EDP Sciences

VL - 43

IS - 4

SP - 757

EP - 784

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, ε,
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.

LA - eng

KW - Iterative initialization;
DAEs;
singular perturbations;
legacy codes;
inertial manifolds.; iterative initialization; daes; legacy codes; recursive projection method; Newton-Krylov generalized minimal residual method

UR - http://eudml.org/doc/250588

ER -

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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).

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