Discrete Kinetic Models and Conservation Laws

Abstract: Classical kinetic theory of gases is based on the Boltzmann equation (BE) which describes the evolution of a system of particles undergoing collisions preserving mass, momentum and energy. Discretization methods have been developed on the idea of replacing the original BE by a finite set of nonlinear hyperbolic PDEs corresponding to the densities linked to a suitable finite set of velocities. One open problem related to the discrete BE is the construction of normal (fulfilling only physical conservation laws) discrete velocity models (DVMs). In many papers on DVMs, authors postulate from the beginning that a finite velocity space with such "good" properties is given, and after this step, they study the discrete BE. Our aim is not to study the equations for DVMs, but to discuss all possible choices of finite phase spaces (sets) satisfying this type of "good" restrictions.We start by introducing the most general class of discrete kinetic models (DKMs) and then, develop a general method for the construction and classification of normal DKMs. We apply this method in the particular cases of DVMs of the inelastic BE (where we show that all normal models can be explicitly described) and elastic BE (where we give a complete classification of normal models up to 9 velocities). 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). We apply this method and obtain SNMs with up to 20 velocities and their spectrum of mass ratio. Finally, we develop a new method that can lead, by symmetric transformations, from a given normal DVM to extended normal DVMs. Many new normal models can be constructed in this way, and we give some examples to illustrate this.