  We investigate various analytical and numerical techniques for the coupling
of nonlinear hyperbolic systems and, in particular, we introduce here an
augmented formulation which allows for the modeling of the dynamics of
interfaces between fluid flows. The main technical difficulty to be overcome
lies in the possible resonance effect when wave speeds coincide and global
hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is
observed for the initial value problem which need to be supplemented with
further admissibility conditions. This first paper is devoted to investigating
these issues in the setting of self-similar vanishing viscosity approximations
to the Riemann problem for general hyperbolic systems. Following earlier works
by Joseph, LeFloch, and Tzavaras, we establish an existence theorem for the
Riemann problem under fairly general structural assumptions on the nonlinear
hyperbolic system and its regularization. Our main contribution consists of
nonlinear wave interaction estimates for solutions which apply to resonant wave
patterns.
