Construction of positive exact (2+1)-dimensional shock wave solutions for two discrete Boltzmann models

Abstract

It is proved that (2+1)-dimensional (spacex, y; timet) positive exact shock wave solutions of two discrete Boltzmann models exist. For each densityN _i, these solutions are linear combinations of three similarity shock waves,N _i =n _0i +∑ _j n _ji /[1+d _j exp(τ _j y+y _j x+ρ _j t)],j=1,2,3. Two models with four independent densities are investigated: the square discrete-velocity Boltzmann model and the model with eight velocities oriented toward the eight corners of a cube.The positivity problem for the densities is nontrivial. Two classes of solutions are considered for which the two first similarity shock wave components depend on only one spatial dimension,γ _j=const·τ _j ,j=1,2. For the positivity, if τ₁τ₂>0, it is sufficient to prove that the 16 asymptotic shock limitsn _0i ,n _0i +n _3i ,∑ ² _j=0 n _ji ,∑ ³ _j=0 n _ji are positive. The density solutions are built up with five arbitrary parameters and we prove that there exist subdomains of the arbitrary parameter space in which the 16 shock limits are positive. We study numerically two explicit shock wave solutions. We are interested in the movement of the shock front when the time is growing and in the possible appearance of bumps. In the space, at intermediate times, these bumps represent populations of particles which are larger than at initial time or at equilibrium time.