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

Vincent Duchêne, Christian Klein

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.

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.

Paper Structure

This paper contains 15 sections, 1 theorem, 35 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. The equations
  4. Outline
  5. Solitary waves
  6. The equations for solitary waves
  7. Numerical construction of solitary waves
  8. Time evolution
  9. The numerical scheme
  10. Validation
  11. Stability of the solitary waves
  12. Emergence of modulated oscillations
  13. Comparison with the Camassa-Holm equation
  14. Near-cavitation initial data
  15. Outlook

Key Result

Proposition 2.1

There exists $(\zeta^{(q)},u^{(q)})_{q>0}$ a one-parameter family of smooth square-integrable functions such that for all $q>0$, $(\zeta_{c_q},u_{c_q}):=(\zeta^{(q)},u^{(q)})$ provides a solitary wave solution to WGN with velocity $c_q>1$, and

Figures (27)

  • Figure 1: Left: solitary wave for the \ref{['WGN']} equations for $c=1.1$ in blue and the \ref{['SGN']} equations for the same velocity in red; right: Fourier coefficients for both solitary waves on the left.
  • Figure 2: Left: solitary wave for the \ref{['WGN']} equations for $c=2$ in blue and the \ref{['SGN']} equations for the same velocity in red; right: Fourier coefficients for the solitary waves on the left.
  • Figure 3: Left: solitary wave for the \ref{['WGN']} equations for $c=20$ in blue and the \ref{['SGN']} equations for the same velocity in red; right: Fourier coefficients on the left.
  • Figure 4: Left: solitary wave for the \ref{['WGN']} equations for $c=100$ in blue and the \ref{['SGN']} equations for the same velocity in red; right: Fourier coefficients on the left.
  • Figure 5: The function $\zeta$ for the solitary wave for the \ref{['WGN']} equations in blue and the \ref{['SGN']} equations for the same velocity in red: left $c=20$, right $c=100$.
  • ...and 22 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Conjecture 2.2
  • Remark 2.3