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.
KeywordsEuler Equation Rarefaction Wave Riemann Problem Contact Discontinuity Riemann Solver
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