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Constraint-Defined Manifolds: a Legacy Code Approach to Low-Dimensional Computation

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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

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References

  1. P. Constantin C. Foias B. Nicolaenko R. Temam (1998) Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations Springer New York

    Google Scholar 

  2. C. Foias M.S. Jolly I.G. Kevrekidis G.R. Sell E.S. Titi (1988) ArticleTitleOn the computation of inertial manifolds Phys. Lett. A 131 433–436 Occurrence Handle89k:65154 Occurrence Handle10.1016/0375-9601(88)90295-2

    Article  MathSciNet  Google Scholar 

  3. C.W. Gear I.G. Kevrekidis (2003) ArticleTitleProjective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum, NEC Technical Report NECI-TR 2001-029 SIAM J. Sci. Comp 24 IssueID4 1091–1106 Occurrence Handle2004c:65065 Occurrence Handle10.1137/S1064827501388157

    Article  MathSciNet  Google Scholar 

  4. C.W. Gear I.G. Kevrekidis (2004) ArticleTitleComputing in the past with forward integration Phy. Lett. A 321 335–343 Occurrence Handle2004m:65106

    MathSciNet  Google Scholar 

  5. Gear C.W., Kaper T.J., Kevrekidis I.G., Zagaris A. Projecting to a slow manifold: singularity perturbed systems and legacy codes, to appear in SIADS; also physics/0405074 at arxiv.org.

  6. Gorban, A. N., and Karlin, I. V. Method of invariant manifolds for chemical kinetics arXiv:cond-mat/0207231.

  7. Gorban, A. N., Karlin, I. V., and Zinoviev, Yu. (2003).Constructive methods of invariant manifolds for kinetic problems, IHES Report M/03/50.

  8. D.A. Jones E.S. Titi (1996) ArticleTitleApproximations of inertial manifolds for dissipative nonlinear equations J. Diff. Equ 127 54–86 Occurrence Handle97g:58154 Occurrence Handle10.1006/jdeq.1996.0061

    Article  MathSciNet  Google Scholar 

  9. M.S. Jolly I.G. Kevrekidis E.S. Titi (1990) ArticleTitleApproximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations Physica D 44 38–60 Occurrence Handle91f:35217 Occurrence Handle10.1016/0167-2789(90)90046-R

    Article  MathSciNet  Google Scholar 

  10. C.T. Kelley (1995) Iterative Methods for Linear and Nonlinear Equations SIAM Publications Philadelphia

    Google Scholar 

  11. S.H. Lam D.A. Goussis (1994) ArticleTitleThe CSP method for simplifying chemical kinetics Int. J. Chem. Kin 26 461–486 Occurrence Handle10.1002/kin.550260408

    Article  Google Scholar 

  12. U. Maas S.B. Pope (1992) Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combustion and Flame 88 239–264

    Google Scholar 

  13. R. Rico-Martinez C.W. Gear I.G. Kevrekidis (2004) ArticleTitle“Coarse projective kMC integration: forward/reverse initial and boundary value problems J. Comp. Phys 196 474–489

    Google Scholar 

  14. J.P. Ryckaert G. Ciccotti H. Berendsen (1977) ArticleTitleNumerical integration of the cartesian equations of motion of a system with constraints: molecular dynamics of N-alkanes J. Comp. Phys 23 327–341

    Google Scholar 

  15. G.M. Shroff H.B. Keller (1993) ArticleTitleStabilization of unstable procedures: A recursive projection method SIAM J. Numer. Anal 30 1099–1120 Occurrence Handle94i:65007 Occurrence Handle10.1137/0730057

    Article  MathSciNet  Google Scholar 

  16. R. Temam (1988) Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer New York

    Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544, USA

    C. William Gear & Ioannis G. Kevrekidis

  2. NEC Research Institute (retired), Princeton, NJ, 08544, USA

    C. William Gear

Authors
  1. C. William Gear
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  2. Ioannis G. Kevrekidis
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Gear, C.W., Kevrekidis, I.G. Constraint-Defined Manifolds: a Legacy Code Approach to Low-Dimensional Computation. J Sci Comput 25, 17–28 (2005). https://doi.org/10.1007/s10915-004-4630-x

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  • DOI: https://doi.org/10.1007/s10915-004-4630-x

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