A variety of phenomena in nature and engineering can be described by the resonant hyperbolic systems, such as the tsunami waves in the ocean, the arterial and venous systems of hemodynamics, the jet engine of aircraft and rocket propulsion systems. The Riemann problem serves as building blocks for the existence and uniqueness of the general Cauchy problem of hyperbolic systems. Hence in this project, we aim to completely solve the Riemann problem for selected resonant hyperbolic systems: the gas dynamic equations in a duct of variable cross–sectional areas and the shallow water equations with a jump in the bottom topography. In the context of the Riemann solutions to the consideration resonant hyperbolic systems, the challenges both for theoretical and numerical studies are here. The first one is to reveal the structure of all resonant waves due to the fact that waves of different families are not well separated and coincide with each other. The second one is to uniformly compute the Riemann problem for any Riemann initial data. The third one is to determine the existence and uniqueness of the weak solutions for the general problem. We solved these problems in [Han et al. 2010, 2012]. The results are summarized in the following.
- The velocity function was introduced to determine the wave curves of the stationary wave. The existence of the stationary waves has been studied for the first time. Specifically, for the gas dynamic equations in a duct of variable cross–sectional areas, on one hand if the duct is expanding monotonic, the stationary wave always exists; on the other hand if the the duct is converging monotonic, the stationary wave exists if and only if the variation of the duct is small enough. To be precise, we defined two critical duct areas to justify that certain stationary waves exist or not. For the shallow water equations, we validated that the water can always spread across a lowered bottom step; But the water can go across an elevated bottom step if and only if a critical step size is larger than the actual jump height of the bottom step. The critical step size is determined by the height and the Froude number of the inflow state. The existence for these two systems provides the methodology for other resonant hyperbolic systems, as well as for the general resonant hyperbolic systems.
- Two basic types of the resonant waves were carefully studied. The first type is due to the coincidence of transonic rarefactions and stationary waves. While the second type is due to the coincidence of stationary waves with 0 speed shocks. The existence and monotonicity of two corresponding composite wave curves were carefully established.
- For simplicity, two combination wave curves in the state space were named L-M and R-M wave curves. They can be classified into different basic cases. The wave configurations and the details of the L-M and R-M wave curves have been completely examined and studied.
- The intersection points of the L–M and R–M curves correspond to the intermediate states of the Riemann solutions. The L-M curve is decreasing and the R-M curve is increasing for most cases. Hence the Riemann solution exists uniquely. However, bifurcations appear in certain cases of L-M and R-M curves. Due to the bifurcations, the L-M and R-M curves are folding in the state space. Therefore, there are more than one intersection points for L–M and R–M curves. In such kind of the case the Riemann solution is nonunique.
- To single out the physically relevant solution among all the possible Riemann solutions, we compared the nonunique Riemann solutions of the gas dynamic equations in ducts with the averaged numerical solutions to compressible axisymmetrical Euler equations computed by the GRP scheme in a cylindrical tube based on unstructured triangle meshes. Here GRP is the abbreviation of the generalized Riemann problem. Andrianov and Warnecke in  suggested using the entropy rate admissibility criterion to rule out the unphysical solutions. However, several examples have been found for which the solution picked out does not have the maximum increase in entropy. Moreover, numerous numerical experiments show that the physically relevant solution is always located on a certain branch of the L–M curves. The bifurcation introduces two additional solutions, but the physical relevant solution is still on the original branch [Han et al. 2013].
In addition, a reduced 3 x 3 mathematical model for the blood flows in medium and large size arteries belongs to the considered resonant hyperbolic systems. The governing system for the blood flows is coupled with tube laws including geometrical and mechanical properties of the blood vessels. The high non-linearity of the tube law is a great challenge for solving the Riemann problem. The present aim of this project is to construct Riemann solutions for subcritical and supercritical Riemann initial data in a uniform manner.