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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)

- S. Ansumali, Iliya V. Karlin
- Physical review. E, Statistical, nonlinear, and…
- 2002

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)

- Alexander N. Gorban, Iliya V. Karlin, Hans Christian Ottinger, Larisa L. Tatarinova
- Physical review. E, Statistical, nonlinear, and…
- 2001

A general method of constructing dissipative equations is developed, following Ehrenfest's idea of coarse graining. The approach resolves the major issue of discrete time coarse graining versus continuous time macroscopic equations. Proof of the H theorem for macroscopic equations is given, several examples supporting the construction are presented, and… (More)

We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in a most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the di erential equation for a… (More)

– 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 expansion of the corresponding local equilibrium is the well-known local… (More)

- Shyam S. Chikatamarla, Iliya V. Karlin
- Physical review. E, Statistical, nonlinear, and…
- 2009

A recently introduced theory of higher-order lattice Boltzmann models [Chikatamarla and Karlin, Phys. Rev. Lett. 97, 190601 (2006)] is elaborated in detail. A general theory of the construction of lattice Boltzmann models as an approximation to the Boltzmann equation is presented. New lattices are found in all three dimensions and are classified according… (More)

Since the invention of the lattice-gas model @1#, latticebased methods for simulations of complex hydrodynamic phenomena received much attention over the past decade. In these methods, hydrodynamic equations are not addressed by a direct discretization procedure, rather, a simple pseudoparticle kinetics is introduced in such a way that the hydrodynamic… (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 grid-based version of MIM is developed.… (More)

where ci is the D-dimensional vector of the ith link, and i 1, . . . , b. The right-hand side of Eq. (1) is the LBGK collision integral Di , the function N eq i is the local equilibrium, and v $ 0 is a dimensionless parameter. As the state of the lattice is updated long enough, the dynamics of Ni becomes governed by macroscopic equations for a finite… (More)

Abstract. Adetailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad’smoment equations. Grad’s systems are considered as theminimal kineticmodels where theChapman-Enskogmethod can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description… (More)