Exact Riemann solutions to two selected resonant hyperbolic systems

We determine completely the Riemann solutions to the gas dynamic equations in a duct with discontinuous varying cross–sectional areas and the shallow water equations with a jump in the bottom topography. In this work the existence of the stationary waves has been studied for the first time. For the gas dynamic equations in a duct with discontinuous cross–sectional areas, the stationary wave always exists for an expanding monotonic duct. But it is complicated in a converging monotonic duct where existence can be found only if the variation of the duct is small enough. To be precise, we define two critical duct areas to justify that certain stationary waves exist or not.