  This series of papers is devoted to the formulation and the approximation of
coupling problems for nonlinear hyperbolic equations. The coupling across an
interface in the physical space is formulated in term of an augmented system of
partial differential equations. In an earlier work, this strategy allowed us to
develop a regularization method based on a thick interface model in one space
variable. In the present paper, we significantly extend this framework and, in
addition, encompass equations in several space variables. This new formulation
includes the coupling of several distinct conservation laws and allows for a
possible covering in space. Our main contributions are, on one hand, the design
and analysis of a well-balanced finite volume method on general triangulations
and, on the other hand, a proof of convergence of this method toward entropy
solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a
single conservation law without coupling). The core of our analysis is, first,
the derivation of entropy inequalities as well as a discrete entropy
dissipation estimate and, second, a proof of convergence toward the entropy
solution of the coupling problem.
