Exact Shock Wave Solutions for an $$8\overrightarrow {{v_i}} $$ Discrete Boltzmann Model with Temperature

Abstract

We study a\( 8\overrightarrow {{v_i}}\) discrete Boltzmann model with speeds \(\sqrt 2\) and velocities along the two medians and the two diagonals of a square. All the conservation laws being satisfied, we can define a nontrivial temperature. We study the physical properties of the shock wave solutions: positivity of the densities, shock front velocity, sound wave velocity, shock velocity, upstream and downstream domains, subsonic and supersonic inequalities. We observe overshoots of the temperature accross the shock.