Basic Types of Coarse-Graining

Gorban, Alexander N.

doi:10.1007/3-540-35888-9_7

Cited by 23 publications

(33 citation statements)

References 119 publications

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“…The idea of artificial partial equilibration steps was proposed by T. and P. Ehrenfest for the foundation of statistical physics [23] and further developed to a general formalism of nonequlibrium thermodynamics [48,18,49]. A review and comparative analysis of different approaches to coarse-graining was published in [44].…”

Section: Discrete Kinetic Models and Lattice Automatamentioning

confidence: 99%

“…This lattice-gas scheme does not coincide with any of the finite difference schemes. Nevertheless, it also models diffusion and, to the first order in τ , D = v 2 τ 2−ω 2ω [98,44]. Now, the area of applications of the cellular and lattice Boltzmann automata is very wide and, in addition to classical fluid dynamics, includes many areas of chemistry [65], models of phase separation [92], dynamics of macromolecules and many other topics.…”

Section: Discrete Kinetic Models and Lattice Automatamentioning

confidence: 99%

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Quasichemical Models of Multicomponent Nonlinear Diffusion

Gorban

Sargsyan

Wahab

2011

Math. Model. Nat. Phenom.

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Abstract. Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusion should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear multicomponent diffusion equations should be ordered and special tools are needed to provide the systematic construction of the nonlinear diffusion equations for multicomponent mixtures with significant interaction between components. We develop an approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism borrowed from chemical kinetics.Chemical kinetics gave rise to very seminal tools for the modeling of processes. This is the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in many areas of science, from particle physics to sociology. In our work we extend the area of applications onto nonlinear multicomponent diffusion.We demonstrate, how the mechanism based approach to multicomponent diffusion can be included into the general thermodynamic framework, and prove the corresponding dissipation inequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law (the generalized Mass Action Law), additional conditions are proved. The cell-jump formalism gives an intuitively clear representation of the elementary transport processes and, at the same time, produces kinetic finite elements, a tool for numerical simulation.

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Section: Discrete Kinetic Models and Lattice Automatamentioning

confidence: 99%

Section: Discrete Kinetic Models and Lattice Automatamentioning

confidence: 99%

Quasichemical Models of Multicomponent Nonlinear Diffusion

Gorban

Sargsyan

Wahab

2011

Math. Model. Nat. Phenom.

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“…We expect that the lattice parameter should partly govern these dynamics and that the 1st order macroscopic dynamics should be governed by the 1st order population functions. In order to find these dynamics we project the microscopic flow (advection) up to the required order, following one time step, onto the invariant manifold up to the same order [7,14].…”

Section: Macroscopic Equationsmentioning

confidence: 99%

Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models

Packwood

Levesley

Gorban

2010

Lecture Notes in Computational Science and Engineering

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Summary. The classical method for deriving the macroscopic dynamics of a lattice Boltzmann system is to use a combination of different approximations and expansions. Usually a Chapman-Enskog analysis is performed, either on the continuous Boltzmann system, or its discrete velocity counterpart. Separately a discrete time approximation is introduced to the discrete velocity Boltzmann system, to achieve a practically useful approximation to the continuous system, for use in computation. Thereafter, with some additional arguments, the dynamics of the Chapman-Enskog expansion are linked to the discrete time system to produce the dynamics of the completely discrete scheme.In this paper we put forward a different route to the macroscopic dynamics. We begin with the system discrete in both velocity space and time. We hypothesize that the alternating steps of advection and relaxation, common to all lattice Boltzmann schemes, give rise to a slow invariant manifold. We perform a time step expansion of the discrete time dynamics using the invariance of the manifold. Finally we calculate the dynamics arising from this system.By choosing the fully discrete scheme as a starting point we avoid mixing approximations and arrive at a general form of the microscopic dynamics up to the second order in the time step. We calculate the macroscopic dynamics of two commonly used lattice schemes up to the first order, and hence find the precise form of the deviation from the Navier-Stokes equations in the dissipative term, arising from the discretization of velocity space.Finally we perform a short wave perturbation on the dynamics of these example systems, to find the necessary conditions for their stability.

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“…Then, after the dynamic motion of the microscopic ensemble under (2), which is conservative, an averaging occurs in the cells, giving rise to an entropy increase. Many other methods in statistical mechanics can be understood as a generalisation of this coarse-graining paradigm [13]. We will be using a modified lattice Boltzmann method to simulate the flow and we will describe this in the more general context of coarse-graining.…”

Section: Coarse-grainingmentioning

confidence: 99%

Stable simulation of fluid flow with high-Reynolds number using Ehrenfests’ steps

2007

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Abstract. The Navier-Stokes equations arise naturally as a result of Ehrenfests' coarse-graining in phase space after a period of free-flight dynamics. This point of view allows for a very flexible approach to the simulation of fluid flow for highReynolds number. We construct regularisers for lattice Boltzmann computational models. These regularisers are based on Ehrenfests' coarse-graining idea and could be applied to schemes with either entropic or non-entropic quasiequilibria. We give a numerical scheme which gives good results for the standard test cases of the shock tube and the flow past a square cylinder.

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Basic Types of Coarse-Graining

Cited by 23 publications

References 119 publications

Quasichemical Models of Multicomponent Nonlinear Diffusion

Quasichemical Models of Multicomponent Nonlinear Diffusion

Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models

Stable simulation of fluid flow with high-Reynolds number using Ehrenfests’ steps

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