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

A numerical scheme for an improved Green-Naghdi model in the Camassa-Holm regime for the propagation of internal waves