Coherence of the iterated valve c-Riemann solver

Determine whether the c-Riemann solver associated with the composition C\circ C of the valve coupling function is coherent, despite the fact that C\circ C is not itself a c-function.

Background

For the pressure-independent one-way valve with hysteresis, the coupling function C maps initial left and right gas states together with the initial valve mode to the corresponding interface traces and updated valve mode. The paper characterizes the incoherence domain of the associated c-Riemann solver: for incoherent data, one iteration of C produces interface data that lie in the coherence domain.

The authors observe that this behavior suggests studying the iterated coupling map C\circ C and conjecturing that the corresponding c-Riemann solver is coherent. However, they also show that C\circ C does not satisfy the definition of a c-function, because its output may violate the admissibility condition defining the left trace set. Thus, establishing coherence for a solver associated with the iterated construction remains an unresolved question under the stated formulation.