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- Alexander N. Gorban, Iliya V Karlin, Springer Berlin, Heidelberg Newyork, Hong Kong, London Milan +53 others
- 2004

Preface This book is about model reduction in kinetics. Is this physics or mathemat-ics? There can be at least four reasonable answers to this question: – It is physics, it is not mathematics; – It is mathematics, it is not physics; – It is both physics and mathematics; – It is neither physics, nor mathematics, it is something else (but what could that… (More)

- I V Karlin, A Ferrante, H C Ottinger
- 1999

PACS. 47.11+j – Computational methods in fluid dynamics. Abstract. – In this letter, we derive entropy functions whose local equilibria are suitable to recover the Navier-Stokes equations in the framework of the Lattice Boltzmann method. For the two-dimensional nine-velocity lattice we demonstrate that such an entropy function is unique, and that the… (More)

Derivation of the lattice Boltzmann method from the continuous kinetic theory [X. He and L. S. Luo, Phys. Rev. E 55, R6333 (1997); X. Shan and X. He, Phys. Rev. Lett. 80, 65 (1998)] is extended in order to obtain boundary conditions for the method. For the model of a diffusively reflecting moving solid wall, the boundary condition for the discrete set of… (More)

- Iliya V Karlin, Alexander N. Gorban, S Succi, V Boffi
- 1998

The entropy maximum approach to constructing equilibria in lattice kinetic equations is revisited. For a suitable entropy function, we derive explicitly the hydrodynamic local equilibrium, prove the H theorem for lattice Bhatnagar-Gross-Krook models, and develop a systematic method to account for additional constraints. Lattice-based simulations of… (More)

A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description… (More)

In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A review of existing alternative methods is… (More)

- Alexander N. Gorban, Iliya V Karlin, Andrei Yu Zinovyev, Andrei Yu, Zinovyev
- 2003

In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A grid-based version of MIM is developed.… (More)

A new approach to the problem of reduced description for Boltzmann-type systems is developed. It involves a direct solution of two main problems: thermodynamici ty and dynamic invariance of reduced description. A universal construction is introduced, which gives a thermodynamic parameterization of an almost arbitrary approximation. Newton-type procedures of… (More)

A new method of successive construction of a solution is developed for problems of strongly nonequilibrium Boltzmann kinetics beyond normal solutions. Firstly, the method provides dynamic equations for any manifold of distributions where one looks for an approximate solution. Secondly, it gives a successive procedure of obtaining corrections to these… (More)

For a one-dimensional benchmark shock tube problem, we implement the lattice Boltzmann method based on the H theorem ͓I. Karlin, A. Ferrante, and H. C. O ¨ ttinger, Europhys. Lett. 47, 182 ͑1999͔͒. Results of simulation demonstrate significant improvement of stability, as compared to realizations without explicit en-tropic estimations.