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

Abstract

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that, in difference to in the continuous case, DVMs can have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species as well as for binary mixtures. For binary mixtures also the concept of supernormal DVMs has been introduced by Bobylev and Vinerean. Supernormal DVMs are defined as normal DVMs such that both restrictions to the different species are normal as DVMs for single species.In this presentation we extend the concept of supernormal DVMs to the case of multicomponent mixtures and introduce it for polyatomic molecules. By polyatomic molecules we mean here that each molecule has one of a finite number of different internal energies, which can change, or not, during a collision. We will present some general algorithms for constructing such models, but also give some concrete examples of such constructions.The two different approaches above can be combined to obtain multicomponent mixtures with a finite number of different internal energies, and then be extended in a natural way to chemical reactions.The DVMs are constructed in such a way that we for the shock-wave problem obtain similar structures as for the classical discrete Boltzmann equation (DBE) for one species, and therefore will be able to apply previously obtained results for the DBE. In fact the DBE becomes a system of ordinary differential equations (dynamical system) and the shock profiles can be seen as heteroclinic orbits connecting two singular points (Maxwellians). The previous results for the DBE then give us the existence of shock profiles for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. For binary mixtures this extension has already been addressed before by the author.