The Riemann Problem for the Euler Equations
In his classical paper of 1959, Godunov  presented a conservative extension of the first-order upwind scheme of Courant, Isaacson and Rees  to non-linear systems of hyperbolic conservation laws. The key ingredient of the scheme is the solution of the Riemann problem. The purpose of this chapter is to provide a detailed presentation of the complete, exact solution to the Riemann problem for the one-dimensional, time-dependent Euler equations for ideal and covolume gases, including vacuum conditions. The methodology can then be applied to other hyperbolic systems.
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