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

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.

To appear in DCDS-B. Figures are reproducible using the Julia package WaterWaves1D at https://github.com/WaterWavesModels/

References in corpus (3)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• A Kinematic Conservation Law in Free Surface Flow
• Global bifurcation of solitary waves for the Whitham equation

Cited by in corpus (2)

• On the blow-up scenario for some modified Serre-Green-Naghdi equations
• Helicity in dispersive fluid mechanics