Boundary layer variational principles: A case study

Grmela, Miroslav; Karlin, Iliya V.; Zmievski, Vladimir B.

doi:10.1103/physreve.66.011201

Cited by 14 publications

(24 citation statements)

References 14 publications

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“…Due to its simplicity, this entropy production principle is especially well suited for the analysis of complex problems. In particular, recently, a version of the entropy production principle was used in the problem of selection of boundary conditions for Grad's moment equations [168,169]. For ideal systems (2.86), as well, as for the Marcelin-De Donder kinetics (2.76) the contribution of the sth reaction toĠ has a particularly simple form:…”

Section: Quasi-steady State Approximationmentioning

confidence: 99%

Invariant Manifolds for Physical and Chemical Kinetics

Gorban¹,

Karlin²

2005

Lecture Notes in Physics

212

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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Section: Quasi-steady State Approximationmentioning

confidence: 99%

Invariant Manifolds for Physical and Chemical Kinetics

Gorban¹,

Karlin²

2005

Lecture Notes in Physics

212

416

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“…There are previous attempts [1,5,10,15] to determine the uncontrollable parameter by some additional physical assumptions or criteria, mostly with unsatisfactory results [5,15] or cumbersome numerical procedures [10]. Unlike those attempts, the present iterative solution does not require any additional assumption.…”

Section: Numerical Iterative Solutionsmentioning

confidence: 91%

“…This problem has been considered in several recent studies [1,5,10,15] of boundary value problems in moment equations with insufficient boundary conditions, by postulating some additional physical assumption or criteria. The present approach does not require any additional physical assumption.…”

Section: Introductionmentioning

confidence: 99%

Weighted iterative solutions of linear differential equations and heat conduction of ideal gases in moment theory

Liu

Vieira

2006

Z. angew. Math. Phys.

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An iterative method is proposed to find a particular solution of a system of linear differential equations, in the form of a fixed-point problem, with no boundary conditions. To circumvent the unboundedness of differential operators, iterative approximation with gradually decreasing weight is used. Conditions for convergence that can easily be checked in numerical iterations are established. Furthermore, for the numerical iterative scheme, uniqueness and stability theorems are proved. These results are applied to heat conduction of ideal gases in moment theory. Mathematics Subject Classification (2000). 65F10, 65N12, 35K05, 74A25.

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“…Due to its simplicity, this entropy production principle is especially well suited for analysis of complex problems. In particular, recently, a version of the entropy production principle was used in the problem of selection of boundary conditions for Grad's moment equations [131,132]. For ideal systems (57), as well, as for the Marcelin-De Donder kinetics (47) the contribution of the sth reaction toĠ has a particularly simple form:…”

Section: Thermodynamic Criteria For Selection Of Important Reactionsmentioning

confidence: 99%

“…The requirement of the entropy growth (universally, for all the reduced models) restricts signiÿcantly the form of projectors (132).…”

Section: Example 4: Nonperturbative Derivation Of Linear Hydrodynamicmentioning

confidence: 99%