Hexagonal discrete Boltzmann models with and without rest particles

Abstract

We consider seven different hexagonal discrete Boltzmann models corresponding to one, two, three, and five hexagons with or without rest particles. In the microscopic collisions the number of particles associated with a given speed is not necessarily conserved, except for two models without rest particles. We compare different behaviors for the macroscopic quantities between models with and without rest particles and when the number of velocities (or hexagons) increases. We study similarity waves with two asymptotic states and consider two classes of solutions at one asymptotic state: either isotropic (densities associated with the same speed are equal) or anisotropic. Two macroscopic quantities seem useful for such studies: internal energy and mass ratio across the asymptotic states, which satisfy a relation deduced from continuous theory. Here we report results for the isotropic solutions, whoch only exist, for both models, in the subdomains where the propagation speed is larger than some well-defined value. Outside these subdomains, modifications occur when the rest particle desity becomes large. For both models we find a monotonic internal energy and subdomains with a mass ratio equal to the one in continuous theory.