  In this paper we introduce a new reformulation of the Green-Naghdi model in
the Camassa-Holm regime for the propagation of internal waves over a flat
topography derived by Duch\^ene, Israwi and Talhouk. These new Green-Naghdi
systems are adapted to improve the frequency dispersion of the original model,
they share the same order of precision as the standard one but have an
appropriate structure which makes them much more suitable for the numerical
resolution. We develop a second order splitting scheme where the hyperbolic
part of the system is treated with a high-order finite volume scheme and the
dispersive part is treated with a finite difference approach. Numerical
simulations are then performed to validate the model and the numerical methods.
