A. V. BOBYLEV and M. C. VINEREAN
Construction and classification of discrete kinetic models without spurious invariants
Received: 9 March 2007
Mathematics Subject Classification (2000): 82C40 - 76P05
Abstract 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 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. The problem has been addressed in the last decade by several authors, in particular by Cercignani, Bobylev, Vedenyapin, Orlov and Cornille. 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 develop 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. We also develop a new method that can lead, by symmetric transformations, from a given normal DVM to extended 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 single gases involved in the mixture, we also obtain normal DVMs). We apply this method and obtain SNMs with up to 20 velocities and their spectrum of mass ratio.