Construction of Discrete Kinetic Models with Given Invariants

Bobylev, A. V.; Vinerean, Mirela

doi:10.1007/s10955-008-9536-9

Cited by 29 publications

(50 citation statements)

References 10 publications

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“…Methods of construction of normal DVMs for monatomic single species and mixtures can be found in e.g. [6,7]. We stress that for normal DVMs we will have exactly s + d + 1 linearly independent collision invariants.…”

Section: Dvms For Mixtures With Polyatomic Moleculesmentioning

confidence: 99%

“…Hence, cf. [7], we need to have d + 1 and d + 2 linearly independent (also with respect to the collisions inside the two species or levels of internal energies) collisions between the two species or levels of internal energies, respectively.…”

Section: Remarkmentioning

confidence: 99%

“…DVMs, without additional collision invariants, for which the collision invariants are linearly independent are called normal. The construction of normal DVMs for single species as well as for binary mixtures has been well studied, see for example [6,7,8] (and references therein), and recently also for multicomponent mixtures [9] and (single species of) polyatomic molecules [10].…”

Section: Introductionmentioning

confidence: 99%

“…The concept of supernormal DVMs for binary mixtures, was introduced in [7] for normal DVMs, such that both the restrictions to the single species also are normal. Later this concept was generalized to the case of multicomponent mixtures in [9] and introduced for polyatomic molecules in [10].…”

Section: Introductionmentioning

confidence: 99%

“…We also conclude that for any finite number of species and any combinations of rational mass ratios and any given (finite number of) multiples of an internal "basic" energy we can construct a supernormal DVM. Our constructed DVMs can always be extended to larger DVMs by the method of one-extensions [11,6,7]. The method of one-extensions is an effective way of creating new normal DVMs out of already existing ones.…”

Section: Introductionmentioning

confidence: 99%

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Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

Bernhoff¹

2016

AIP Conference Proceedings

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Abstract. An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that, in difference to in the continuous case, DVMs can have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species as well as for binary mixtures. For binary mixtures also the concept of supernormal DVMs has been introduced by Bobylev and Vinerean. Supernormal DVMs are defined as normal DVMs such that both restrictions to the different species are normal as DVMs for single species.In this presentation we extend the concept of supernormal DVMs to the case of multicomponent mixtures and introduce it for polyatomic molecules. By polyatomic molecules we mean here that each molecule has one of a finite number of different internal energies, which can change, or not, during a collision. We will present some general algorithms for constructing such models, but also give some concrete examples of such constructions.The two different approaches above can be combined to obtain multicomponent mixtures with a finite number of different internal energies, and then be extended in a natural way to chemical reactions.The DVMs are constructed in such a way that we for the shock-wave problem obtain similar structures as for the classical discrete Boltzmann equation (DBE) for one species, and therefore will be able to apply previously obtained results for the DBE. In fact the DBE becomes a system of ordinary differential equations (dynamical system) and the shock profiles can be seen as heteroclinic orbits connecting two singular points (Maxwellians). The previous results for the DBE then give us the existence of shock profiles for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case.

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Section: Dvms For Mixtures With Polyatomic Moleculesmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

Bernhoff¹

2016

AIP Conference Proceedings

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Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas

Bernhoff

2012

J Stat Phys

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We consider a non-linear half-space problem related to the condensation problem for the discrete Boltzmann equation and extend some known results for a single-component gas to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian at infinity, as the non-condensable gas tends to zero at infinity. We assume that the vapor is completely absorbed and that the non-condensable gas is diffusively reflected at the condensed phase and that the vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution, needed for the existence of a unique solution of the problem are investigated. We also find exact solvability conditions and solutions for a simplified six+four-velocity model, as the given distribution is a Maxwellian at rest, and study a simplified twelve+six-velocity model.Keywords Boltzmann equation · boundary layers · discrete velocity models · half-space problem · non-condensable gasIn this paper we consider the condensation problem for a single-component gas or vapor when a non-condensable gas is present [15]. Formulation and motivation of the problem can be found in [15]. The vapor is assumed to tend to an assigned Maxwellian M A ∞ , with a flow velocity towards the condensed phase, at infinity, while the non-condensable gas tends to zero at infinity.

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Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation

Bernhoff

2015

J Stat Phys

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We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation. Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calculated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.Keywords Bose-Einstein condensate · Low temperature kinetics · Discrete kinetic equation · Milne problem · Kramer problem IntroductionHalf-space problems have an important role in the study of the asymptotic behavior of the solutions of boundary value problems of kinetic equations for small Knudsen numbers [4,15,16,26,27]. In this paper we study half-space problems related to a quantum kinetic equation [23,33], for the distribution function of excited atoms interacting with a Bose-Einstein condensate. Motivated by the work

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Construction of Discrete Kinetic Models with Given Invariants

Cited by 29 publications

References 10 publications

Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas

Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation

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