SHOCK WAVES FOR THE TWO SPEEDS 8V_i , 14V_i AND 24V_i DISCRETE BOLTZMANN MODELS WITH TEMPERATURE

For the discrete Boltzmann models in R^q+1 q=1,2,3, with two speeds ,1 or 2 we consider the shock waves along an axis for both the square 8v_i, cubic 14v_i and hypercubic 24v_i models. They satisfy two classes of q-dependent non linear equations and the main difference comes from the projections of the speeds along the axis which are either 1 or 1,2. First we compare the two methods of shock waves solutions: either the Rankine-Hugoniot relations for traveling waves or the similarity solutions. The first method does not always lead to positive cross-sections. Second we study the local entropy and temperature overshoots across the shock.