Résumé
We derive transmission operators for coupling linear Green-Naghdi equations (LGNE) with linear shallow water equations (LSWE) --the heterogeneous case -- or for coupling LGNE with LGNE --the homogeneous case. We derive them from a domain decomposition method (Neumann-Dirichlet) of the linear Euler equations by applying the same vertical-averaging process and truncation of the asymptotic expansion of the velocity field used in the derivation of the equations. We find that the new asymptotic transmision conditions also correspond to Neumann and Dirichlet operators. In the homogeneous case the method has the same convergence condition as the parent domain decomposition method but leads to a solution that is different from the monodomain solution due to an $O(\Delta x)$ term. In the heterogeneous case the Neumann-Dirichlet operators translate into a simple interpolation across the interface, with an extra $O(\Delta x^2)$ term. We show numerically that in this case the method introduces oscillations whose amplitude grows as the mesh is refined, thus leading to an unstable scheme.