Positive exact 2+1 dimensional solutions for discrete Boltzmann models

We have constructed 2+1 dimensional exact solutions (space x, y time t) for three models: i) the square velocity model and the cubic model (with eight velocities oriented towards the eight corners) which satisfy the same type of nonlinear equations, ii) the Broadwell model. The main difficulty is the positivity condition Ni>0 for the densities Ni. We have obtained three classes of exact solutions: i) densities relaxing towards noniform Maxwellians ii) shock wave iii) semi-periodic solutions