Coherence of Coupling Conditions for the Isothermal Euler System

We consider an isothermal flow through two pipes. At the junction, the flow is possibly modified by some devices, such as valves, compressors, and so on, or by the geometry of the junction; coupling conditions between the traces of the flow must be given. We first provide a general framework to model this situation by means of constrained Riemann problems and provide some theoretical results. A key issue for both the validity of a coupling model and the robustness of numerical schemes to find solutions is whether the coupling Riemann solver is coherent. This property implies that applying the coupling Riemann solver to the traces at the junction of a coupling solution results in finding the same solution locally. We also give theoretical results for coherence. Then, we consider several couplings; we discuss the uniqueness of the corresponding solvers and, in particular, their coherence. Surprisingly, some solvers of wide use are proven not to be uniquely defined, and others are not coherent. We present numerical examples to illustrate this property.