Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model

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.