Legendre integrators, post-processing and quasiequilibrium

Gorban, Alexander N.; Gorban, P.A.; Karlin, Iliya V.

doi:10.1016/j.jnnfm.2003.12.005

Cited by 11 publications

(15 citation statements)

References 57 publications

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“…It is not the whole truth, even for the FENE-P equation, as it was shown in [109,369]. The FokkerPlanck equation describes the shape of a probability cloud in the space of conformations.…”

Section: Post-navier-stokes Equationsmentioning

confidence: 99%

“…In most of the works (of us and of other people on similar problems), analytic forms were required to represent manifolds (see, however, the method of Legendre integrators [254,266,369]). However, in order to construct manifolds of a relatively low dimension, grid-based representations of manifolds become a relevant option.…”

Section: Invariant Gridsmentioning

confidence: 99%

“…In accordance with [369], the ansatz for Ψ can be suggested in the following form: To build the thermodynamic projector on the manifold (13.113), the thermodynamic Lyapunov function is necessary. It is necessary to emphasize that equations (13.104) are nonlinear.…”

Section: Two-peak Approximation For Polymer Stretching In Flow and Exmentioning

confidence: 99%

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Invariant Manifolds for Physical and Chemical Kinetics

Gorban¹,

Karlin²

2005

Lecture Notes in Physics

210

416

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Of course, it is physics. Model reduction in kinetics requires physical concepts and structures; it is impossible to make an expedient reduction of a kinetic model without thermodynamics, for example. The entropy, the Legendre transformation generated by the entropy, and the Riemann structure defined by the second differential of the entropy provide the elementary geometrical basis for the first approximation. The physical sense of the models gives many hints for their further processing. So, it is not mathematics; we care about the physical sense more than about rigorous proofs. We should deal with equations even in the absence of theorems about existence and uniqueness of solutions. Mathematics assimilates the physical notions with a considerable delay in time, but any such an assimilation leads to further insights. 1But, without doubt, it is mathematics. The story about invariant manifolds for differential equations began inside mathematics. The first significant steps were taken by two great mathematicians, A.M. Lyapunov and H. Poincaré, at the end of the XIXth century. Then N.M. Krylov and N.N. Bogolyubov, A.N. Kolmogorov, V.I. Arnold and J. Moser, J.E. Marsden, M.I. Vishik, R. Temam, and many other mathematicians developed this field of science, and many elegant theorems and useful methods were created. This is not only pure mathematics, the wide field of applications was developed too, from hydrodynamics to process engineering and control theory and methods. This is pure and applied dynamics. The language of model reduction, the basic notions that we use, the theorems and methods, all this either came from 1 The closest example: after mathematicians discovered how the entropy functional may be important for the theory of the Boltzmann equation, then they proved the existence theorem (P.L. Lions and R. DiPerna, this work was awarded the Fields medal in 1994). VIIIPreface pure and applied dynamics directly, or bears the visible imprint of its ideas and methods. Maybe the book presents a specific chapter on this subject? But, of course, the problems came from physics, from engineering. Maybe it is both physics and mathematics? Or perhaps it is something different, but what can it be? It is not so easy to answer the question, what is the subject of our book, even for the authors. But we can say what we want it to be. We want it to be a special "meeting point" of pure and applied dynamics, of physics, and of engineering sciences. This meeting point has a sufficient number of specific problems, methods and results to deserve a special name. We propose the name Model Engineering. As long as it is engineering, it is synthetic subject: if it is possible to prove something exactly, this is great, and we should follow this possibility, but if the physical sense gives us a seminal hint, well, we should use it even if the rigorous foundations are far from complete. The result is the model that works. In this enormous field of intellectual activity our book tends to be in the theoretical corner; we focus our study on c...

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“…It is not the whole truth, even for the FENE-P equation, as it was shown in [109,369]. The FokkerPlanck equation describes the shape of a probability cloud in the space of conformations.…”

Section: Post-navier-stokes Equationsmentioning

confidence: 99%

Section: Invariant Gridsmentioning

confidence: 99%

See 1 more Smart Citation

Invariant Manifolds for Physical and Chemical Kinetics

Gorban¹,

Karlin²

2005

Lecture Notes in Physics

210

416

View full text Add to dashboard Cite

show abstract

“…It is not the whole truth, even for the FENE-P equation, as it was shown in Refs. [84,264]. The Fokker-Planck equation describes the shape of a probability cloud in the space of conformations.…”

Section: Polymodal Polyhedron and Molecular Individualismmentioning

confidence: 99%

Constructive methods of invariant manifolds for kinetic problems

2004

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“…In most of the works (of us and of other people on similar problems), analytic forms were required to represent manifolds (see, however, the method of Legendre integrators [11][12][13] ). However, in order to construct manifolds of a relatively low dimension, grid-based representations of manifolds become a relevant option [8] .…”

Section: Invariant Gridsmentioning

confidence: 99%

Invariant Grids: Method of Complexity Reduction in Reaction Networks

2004

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Complexity in the description of big chemical reaction networks has both structural (number of species and reactions) and temporal (very different reaction rates) aspects. A consistent way to make model reduction is to construct the invariant manifold which describes the asymptotic system behaviour. In this paper we present a discrete analog of this object: an invariant grid. The invariant grid is introduced independently from the invariant manifold notion and can serve to represent the dynamic system behaviour as well as to approximate the invariant manifold after refinement. The method is designed for pure dissipative systems and widely uses their thermodynamic properties but allows also generalizations for some classes of open systems. The method is illustrated by two examples: the simplest catalytic reaction (Michaelis-Menten mechanism) and the hydrogen oxidation.

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Legendre integrators, post-processing and quasiequilibrium

Cited by 11 publications

References 57 publications

Invariant Manifolds for Physical and Chemical Kinetics

Invariant Manifolds for Physical and Chemical Kinetics

Constructive methods of invariant manifolds for kinetic problems

Invariant Grids: Method of Complexity Reduction in Reaction Networks

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