Numerical Approximations of Pressureless and Isothermal Gas Dynamics

We study several schemes of rst or second-order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations. The pressureless gas system is weakly hyperbolic, giving rise to the formation of density concentrations known as delta-shocks. For the isothermal gas system, the innnite speed of expansion into vacuum leads to zero timestep in the Godunov method based on exact Riemann solver. The schemes we consider are able to bypass these diiculties. They are proved to satisfy positiveness of density and discrete entropy inequalities, to capture the delta-shocks and treat data with vacuum.