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 Favrie and Gavrilyuk (2017 Nonlinearity 30 2718–36) 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 sufficiently regular solution to 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.