9Vi discrete Boltzmann model with multiple collisions

Abstract

URL: http://www-spht.cea.fr/articles/t93/055We study a hierarchy of discrete Boltzmann models (DBMs) with speeds $0,1,\sqrt 2$ when, in addition to binary collisions, ternary and quaternary multiple collisions are included: i) the square $9v_i$ model, ii) an associated three dimensional $15v_i$ model. Firstly we find, for shock waves, that the two equilibrium states are the same for binary alone collisions or not. We deduce, from the H-Theorem, a criterion for any multiple collision term. Secondly, from the knowledge of only the two equilibrium states associated to ``shock profiles" solutions we can predict whether or not overshoots for the ratios $P/M$ ($P$ for pressure, $M$ for mass and $P/M$ for internal energy) are possible. In the arbitrary parameter space of the two equilibrium states we are able to predict the subdomains where both overshoots can occur or not and the strength of the effect. These subdomains are characterized by the singularities of the propagation speed $\zeta$. Comparing with the square $8v_i$ model, without rest particles, a great difference occurs for $\zeta \simeq 0$. These predictions are independent of the fact that multiple collisions are present or not and can be generalized to any other DBM. Finally we construct exact similarity shock waves when ternary collisions are present, observe thinner shock profiles and verify the previous predictions on the $P/M$ behaviours