Discrete Kinetic Models and Conservation Laws

Prof. Alexander Bobylev

Abstract: Discrete Kinetic Models (DKMs) describe an evolution of a media consisting of enormous number of particles (gases, plasmas, etc.). DKMs are interesting from purely mathemetical point of view as discrete models of more complicated "continuous" kinetic equations (this is a reason why Carleman introduces the first such model in 1932), and also they are used now in many applied numerical works. The main idea of DKMs is to replace the "continuous" phase space R_n by a finite set of phase points. This re-placement is, however, not an easy problem since very often we automatically get so-called spurious invariants which are not related to any physical conservation laws. This problem was raised already in 1975 by R.Gatignol and still remains, generally speaking, unsolved.

The talk is devoted to a recently developed approach to this problem. It is shown that the problem can be expressed in terms of functional equations on finite set and that it can be, in principle, solved in the most general case with surprisingly simple mathematical tools. We show that, in most practically interesting cases, there exist just a finite number of different classes of "good" DKMs and they all can be constructed with a help of computer. The results are illustrated by a complete list of "good" DKMs of the classical Boltzmann equation on the plane with small number of velocities N<10. Numerical calculations, however, become difficult for large numbers N (the number of operations grows at least as (N!)/2 ).