  We continue our analysis of the coupling between nonlinear hyperbolic
problems across possibly resonant interfaces. In the first two parts of this
series, we introduced a new framework for coupling problems which is based on
the so-called thin interface model and uses an augmented formulation and an
additional unknown for the interface location; this framework has the advantage
of avoiding any explicit modeling of the interface structure. In the present
paper, we pursue our investigation of the augmented formulation and we
introduce a new coupling framework which is now based on the so-called thick
interface model. For scalar nonlinear hyperbolic equations in one space
variable, we observe that the Cauchy problem is well-posed. Then, our main
achievement in the present paper is the design of a new well-balanced finite
volume scheme which is adapted to the thick interface model, together with a
proof of its convergence toward the unique entropy solution (for a broad class
of nonlinear hyperbolic equations). Due to the presence of a possibly resonant
interface, the standard technique based on a total variation estimate does not
apply, and DiPerna's uniqueness theorem must be used. Following a method
proposed by Coquel and LeFloch, our proof relies on discrete entropy
inequalities for the coupling problem and an estimate of the discrete entropy
dissipation in the proposed scheme.
