Abstract: The quasi-one-dimensional Euler equations in a duct of variable cross-section are probably one of the most simplest non-conservative systems. We consider the Riemann problem for it and discuss its properties. In particular, for some initial conditions, the solution to the Riemann problem appears to be non-unique. In order to rule out the non-physical solutions, we provide 2D computations of the Euler equations in a duct of corresponding geometry and compare it with the 1D results. Then, the physically relevant 1D solutions satisfy a kind of entropy rate admissibility criterion. Finally, we present a procedure for finding an exact solution to the Riemann problem and construct a Godunov-type method on its basis.
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