Construction of Discrete Kinetic Models with Given Invariants |
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Authors: | A V Bobylev M C Vinerean |
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Institution: | (1) Department of Mathematics, Karlstad University, 65188 Karlstad, Sweden |
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Abstract: | We consider the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem
was first stated by Gatignol in connection with discrete models of the Boltzmann equation (BE) and it has been addressed in
the last decade by several authors. Even though a practical criterion for the non-existence of spurious conservation laws
has been devised, and a method for enlarging existing physical models by new velocity points without adding non-physical invariants
has been proposed, 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 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 classes of DKMs. We apply the general method in the particular cases of
discrete velocity models (DVMs) of the inelastic BE and elastic BE. 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 single gases involved in the mixture, we also obtain normal DVMs). |
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Keywords: | Boltzmann equation Discrete kinetic models Conservation laws Collision invariants |
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