Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

TL;DR

This paper numerically compares the Serre-Green-Naghdi equations with a fully dispersive counterpart, analyzing solitary wave stability, oscillation emergence, and solution blow-up using spectral and Runge-Kutta methods.

Contribution

It introduces a numerical framework for studying fully dispersive wave equations and compares their solutions to classical models, highlighting stability and blow-up phenomena.

Findings

Stable solitary waves for both models identified.

Emergence of modulated oscillations observed.

Conditions leading to solution blow-up analyzed.

Abstract

We perform numerical experiments on the Serre-Green-Naghdi (SGN) equations and a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the emergence of modulated oscillations and the possibility of a blow-up of solutions in various situations is investigated. We argue that a simple numerical scheme based on a Fourier spectral method combined with the Krylov subspace iterative technique GMRES to address the elliptic problem and a fourth order explicit Runge-Kutta scheme in time allows to address efficiently even computationally challenging problems.