The closure problem in discrete kinetics

Abstract

The purpose of this article is to introduce the concept of possible implementation of continuum mechanics models based on kinetic theory. The fundamental equation of rarefied gas dynamics – the Boltzmann equation – is an integro-differential equation, extremely complex when its actual resolution is necessary. The mismatch between the rightand left-hand sides of the Boltzmann kinetic equation is well known, especially in numerical experiments. The imperfection of the Boltzmann kinetic equation led to the need to construct so-called discrete kinetic equations. Several simplified models have been proposed, including discrete kinetic equations with a finite number of group velocities. These models have interesting conceptual and mathematical features. Invariant solutions in discrete kinetics are states or relations that are preserved over time during the evolution of a system described by discrete kinetic equations. Their role is fundamental both in theoretical analysis and in applications. The invariants reflect the fundamental conservation laws for the number of particles (the mass balance), momentum, energy, charge, etc. In discrete models, this ensures that the numerical scheme does not artificially generate or destroy physical quantities. We consider the problem of closing moment chains constructed with invariant solutions and investigate problems of stabilizing periodic perturbations of the equilibrium position for a one-dimensional 6-velocity model. An exponentially fast stabilization of periodic perturbations of the equilibrium position to a traveling wave is established (with general periodic initial perturbations).