We study several schemes of first 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 infinite 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 difficulties.
They are proved to satisfy positiveness
of density and discrete entropy inequalities,
to capture the delta-shocks and treat data with vacuum.
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