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General Methods of the Construction of Discrete Kinetic Models with Given Conservation Laws
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
2007 (English)Conference paper, Published paper (Refereed)
Abstract [en]

In the present work we consider the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem was first stated by R. Gatignol [1] in connection with discrete models of the Boltzmann equation (BE), when it became clear that the velocity discretization can lead to equations with spurious conservation laws (not linear combinations of physical invariants). The problem has been addressed in the last decade by several authors, in particular by Cercignani, Bobylev, Vedenyapin, and Cornille. Even though a practical criterion for the non-existence of spurious conservation laws has been devised [2], and a method for enlarging existing physical models by new velocity points without adding non-physical invariants has been proposed [3], a general algorithm for the construction of all normal (physical) discrete models with assigned conservation laws, in any dimension and for any number of points, is still lacking in the literature. We develop such a general algorithm in the present work.

We introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal DKMs. In particular, it is proved that for any given dimension d 2 and for any sufficiently large number N of velocities (for example, N 6 for the planar case d = 2) there exists just a finite number of distinct (non-equivalent) classes of DKMs. We apply the general method in the particular cases of discrete velocity models (DVMs) of the inelastic BE and elastic BE. In the first case, we show that all normal models can be explicitly described. In the second case, we give a complete classification of normal models up to 9 velocities and show that the extension method [3], does not lead to all normal DVMs.

Using our general approach to DKMs and our results on normal DVMs for a single gas, we develop a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures. We call such models supernormal models (SNMs) (they have the property that by isolating the velocities of one-kind particles of the single gases involved in the mixture, the corresponding discrete models for a single gas are also normal models). We apply this method and obtain SNMs with up to 20 velocities and their spectrum of mass ratio.

Finally, we develop a new method that can lead, by symmetric transformations, from a given normal DVM to extended normal DVMs.



ACKNOWLEDGEMENTS. The support by the grant 2003-5357 from Swedish Research Council for both authors is gratefully acknowledged.



REFERENCES

[1] R. Gatignol, Théorie Cinétique des Gaz à Répartition Discrète de Vitesses, Springer-Verlag, New-York, 1975

[2] V. V. Vedenyapin, Y. N. Orlov, Teoret. and Math. Phys., 121, 1516-1523 (1999)

[3] A. V. Bobylev, C. Cercignani, J. Statist. Phys., 97, 677-686 (1999)

Place, publisher, year, edition, pages
2007.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-19741OAI: oai:DiVA.org:kau-19741DiVA, id: diva2:593392
Conference
Rarefied Gas Dynamics, M.S.Ivanov and A.K.Rebrov, eds., Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 2007, pp. 209-214
Available from: 2013-01-21 Created: 2013-01-21 Last updated: 2013-01-21

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Bobylev, AlexanderVinerean-Bernhoff, Mirela

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