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The paper introduces into the theory of traces [1] which provides an intermediate description between fluid dynamics and mesoscopic theory. It is derived through the projection of the local kinetic equations onto the space of closure moments (presure components, heat flux). This approach allows a comparison between different kinetic models (in particular Boltzmann and BGK) and to understand some "ghost effect" [2] of kinetic theory. [ABSTRACT FROM AUTHOR]

In [2, 3], a framework for a calculus with Discrete Velocity Models (DVM) has been derived. The rotatonal symmetry of the discrete velocities can be modelled algebraically by the action of the cyclic groupC4 - or including reflections - of the dihedral group D4. Taking this point of view, the linearized collision operator can be represented in a compact form as a matrix of elements in the group algebra. Or in other words, by choosing a special numbering it exhibits a certain block structure which lets it appear as a matrix with entries in a certain polynomial ring. A convenient way for approaching such a structure is the use of a computer algebra system able to treat these (predefined) algebraic structures. We used the computer algebra system FriCAS/AXIOM [4, 5] for the generation of the velocity and the collision sets and for the analysis of the structure of the collision operator. Concerning the fluid dynamic limit, the system provides the characterization of sets of collisions and their contribution to the flow parameters. It allows the design of rotationally invariant symmetric models for prescribed Prandtl numbers. The implementation in FriCAS/AXIOM is explained and its results for a 25-velocity model are presented. [ABSTRACT FROM AUTHOR]

It is a well-known fact that steady transport operators based on the Boltzmann collision operator and linearized around a Maxwellian at zero bulk velocity exhibit a typical structure concerning the null space (see e.g. [1]). Introducing a small bulk velocity breaks up this structure and gives rise to a certain bifurcation phenomenon which can be described explicitly in terms of the characteristic quantities of the collision model. Based on Discrete Velocity Models (DVM) we combine this feature with an appropriate scaling which has proven useful in a number of models for diffusion phenomena (e.g. [2]). As a result, this approach provides explicit formulas which may serve as closure relations for the moment hierarchy in the macroscopic limit. As an example, the same principle can be readily applied to an evaporation condensation problem for a binary gas mixture which was discussed in the literature [3,4] and which under the classical asymptotic analysis approach leads to a "ghost effect" which is hard to interpret. The method at hand, however, resolves this seeming paradox. [ABSTRACT FROM AUTHOR]