Quasi-equilibrium closure hierarchies for the Boltzmann equation

Gorban, Alexander N.; Karlin, Iliya V.

doi:10.1016/j.physa.2005.07.016

Cited by 29 publications

(35 citation statements)

References 51 publications

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“…Still, the number is large and we recently developed a code which allows to evaluate ͑23͒ using a symbolic programming language. 20 The range of applicability of Grad's distribution ͑21͒ has been discussed, [31][32][33][34] and specifically also in the vicinity of boundaries. 35 The equilibrium Maxwell distribution is a trivial special case of ͑21͒.…”

Section: Compact Results For Grad's 13 Moment Expansionmentioning

confidence: 99%

Unifying kinetic approach to phoretic forces and torques onto moving and rotating convex particles

Kröger

Hütter

2006

The Journal of Chemical Physics

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We derive general expressions and present several examples for the phoretic forces and torques acting on a translationally moving and rotating convex tracer particle, usually a submicrosized aerosol particle, assumed to be small compared to the mean free path of the surrounding nonequilibrium gas. Point of departure is an expression of the stress tensor in terms of half-sphere integrals to be evaluated with the inhomogeneous velocity distribution function of the surrounding gas, more specifically, its approximation in terms of a finite number of moments. A worked out example covers Grad’s 13 moment approximation in and out of the so-called hydrodynamic approximation. We implement an accommodation coefficient characterizing the collision process in order to derive the tracer particle’s complete equations of motion in a form which offers the possibility for immediate numerical implementation, and the analytical exploration of the effect of particle shape and size on phoretic drift velocity. Explicit expressions for axisymmetric, spherical, cylindrical, and also cuboidal particles are presented, discussed, and successfully compared with the rarely available, previously treated special cases, thus supporting the unifying approach presented in this manuscript.

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Section: Compact Results For Grad's 13 Moment Expansionmentioning

confidence: 99%

Unifying kinetic approach to phoretic forces and torques onto moving and rotating convex particles

Kröger

Hütter

2006

The Journal of Chemical Physics

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“…It was found in [224] that the Grad distributions are linearized versions of appropriate quasiequilibrium approximations (see also [230,233,234]). A method which treats fluxes (e.g.…”

Section: Difficulties Of Classical Methods Of the Boltzmann Equation mentioning

confidence: 99%

“…The maximum entropy principle was applied to the description the universal dependence the three-particle distribution function F 3 on the two-particle distribution function F 2 in classical systems with binary interactions [229]. For a discussion the quasiequilibrium moment closure hierarchies for the Boltzmann equation [224] see the papers [230,233,234]. A very general discussion of the maximum entropy principle with applications to dissipative kinetics is given in the review [231].…”

Section: Entropy and Quasiequilibriummentioning

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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“…For an appropriate initial condition from q (not sufficiently close to q 0 ), two steps of LBGK with I β 0 give the same second-order accuracy as (29). But a long chain of such steps can lead far from the quasiequilibrium manifold and even from q.…”

Section: Introductionmentioning

confidence: 97%

Stability and stabilization of the lattice Boltzmann method

2007

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We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager-Gross-Krook method (LBGK).The LBGK scheme can be recognised as a discrete dynamical system generated by free-flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modelling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface -the invariant film (up to second-order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free-flight.The main instability mechanisms are identified. The simplest recipes for stabilisation add no artificial dissipation (up to second-order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blow-up. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a 1D shock tube and the unsteady 2D-flow around a square-cylinder up to Reynolds number O(10000).

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Quasi-equilibrium closure hierarchies for the Boltzmann equation

Cited by 29 publications

References 51 publications

Unifying kinetic approach to phoretic forces and torques onto moving and rotating convex particles

Unifying kinetic approach to phoretic forces and torques onto moving and rotating convex particles

Invariant Manifolds for Physical and Chemical Kinetics

Stability and stabilization of the lattice Boltzmann method

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