Constraint-defined manifolds: A legacy code approach to low-dimensional computation
C. William Gear, Ioannis G. Kevrekidis
If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of "slow" variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.
Journal of Scientific Computing
Inertial manifolds
Stiff equations
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Slow Manifold Mathematics
Simulators Engineering & Materials Science
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Differential equations Engineering & Materials Science
Inertial Manifolds Mathematics
Parameterise Mathematics
Stationary point Mathematics
Manipulation Mathematics
Gear, C. W., & Kevrekidis, I. G. (2005). Constraint-defined manifolds: A legacy code approach to low-dimensional computation. Journal of Scientific Computing, 25(1), 17-28. https://doi.org/10.1007/s10915-004-4630-x
Gear, C. William ; Kevrekidis, Ioannis G. / Constraint-defined manifolds : A legacy code approach to low-dimensional computation. In: Journal of Scientific Computing. 2005 ; Vol. 25, No. 1. pp. 17-28.
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title = "Constraint-defined manifolds: A legacy code approach to low-dimensional computation",
abstract = "If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of {"}slow{"} variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.",
keywords = "Differential equations, Inertial manifolds, Stiff equations",
author = "Gear, {C. William} and Kevrekidis, {Ioannis G.}",
journal = "Journal of Scientific Computing",
Gear, CW & Kevrekidis, IG 2005, 'Constraint-defined manifolds: A legacy code approach to low-dimensional computation', Journal of Scientific Computing, vol. 25, no. 1, pp. 17-28. https://doi.org/10.1007/s10915-004-4630-x
Constraint-defined manifolds : A legacy code approach to low-dimensional computation. / Gear, C. William; Kevrekidis, Ioannis G.
In: Journal of Scientific Computing, Vol. 25, No. 1, 10.2005, p. 17-28.
T1 - Constraint-defined manifolds
T2 - A legacy code approach to low-dimensional computation
AU - Gear, C. William
AU - Kevrekidis, Ioannis G.
N2 - If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of "slow" variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.
AB - If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of "slow" variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.
KW - Differential equations
KW - Inertial manifolds
KW - Stiff equations
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
Gear CW, Kevrekidis IG. Constraint-defined manifolds: A legacy code approach to low-dimensional computation. Journal of Scientific Computing. 2005 Oct;25(1):17-28. https://doi.org/10.1007/s10915-004-4630-x