Discrete velocity models of the Boltzmann equation and conservation laws

Abstract

The known nonlinear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim -Uehling -Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann -type kinetic equation that depends on a function of four real variables F (x, y; v, w). The function F is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the above mentioned kinetic equations correspond to different forms of the function (polynomial) F . Then the problem of discretization of the general Boltzmann -type kinetic equation is considered on the basis of ideas which are similar to those used for construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to Boltzmann H-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of approximation of the Boltzmann-type equation by its discrete models and the problem of construction of normal discrete models are also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows to read the paper without any preliminary knowledge in kinetic theory.

Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation

“…Construction of normal discrete kinetic models and especially DVMs have been extensively studied, see for example [10,12,13] and references therein. A Maxwellian distribution or just Maxwellian is on the form…”

Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

“…DVMs, without additional collision invariants, for which the collision invariants are linearly independent are called normal. The construction of normal DVMs for single species as well as for binary mixtures has been well studied, see for example [6,7,8] (and references therein), and recently also for multicomponent mixtures [9] and (single species of) polyatomic molecules [10].…”

Discrete Velocity Models for Polyatomic Molecules Without Nonphysical Collision Invariants

“…Intuitively, a set of collisions is linearly dependent if one of them can be obtained by a combination of (some of) the other collisions (including corresponding reverse collisions), and correspondingly linearly independent if this is not the case. More formally, each collision can be represented by an n-dimensional vector with 0, −1, and 1 as the only coordinates, see, e.g., [16,18], in the way that collision (7) is represented by a vector (with non-zero elements at the positions i, j, k, and l)…”

“…DVMs, without additional collision invariants, for which the collision invariants are linearly independent, are called normal. The construction of normal DVMs for single species as well as for binary mixtures has been well studied, see for example [15,16,18,23,24,[35][36][37], and recently also for multicomponent mixtures [10]. We like to point out, that in [16] main ideas of a general approach to the construction of normal discrete kinetic models, including a brief discussion on the application to DVMs for inelastic collisions, is presented.…”