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

TL;DR

This paper rigorously proves that the Favrie-Gavrilyuk approximation converges to the Serre-Green-Naghdi model as a parameter tends to infinity, providing a solid mathematical foundation for this numerical approach in shallow-water wave modeling.

Contribution

It offers the first rigorous justification of the Favrie-Gavrilyuk approximation as a limit of the Green-Naghdi system, addressing a key gap in the theoretical understanding.

Findings

Quantitative convergence estimates established

Favrie-Gavrilyuk solutions approximate Green-Naghdi solutions as parameter increases

Addresses low Mach number limit complexities

Abstract

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.