  The (Serre-)Green-Naghdi system is a non-hydrostatic model for the
propagation of surface gravity waves in the shallow-water regime. Recently ,
Favrie and Gavrilyuk proposed in [Nonlinearity, 30(7) (2017)] an efficient way
of numerically computing approximate solutions to the Green-Naghdi system. The
approximate solutions are obtained through solutions of an augmented
quasilinear system of balance laws, depending on a parameter. In this work, we
provide quantitative estimates showing that any regular solution of the
Green-Naghdi system is the limit of solutions to the Favrie-Gavrilyuk system as
the parameter goes to infinity, provided the initial data of the additional
unknowns is well-chosen. The problem is therefore a singular limit related to
low Mach number limits with additional difficulties stemming from the fact that
both order-zero and order-one singular components are involved.
