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Numerical study of the Serre-Green-Naghdi equations in 2D

S. Gavrilyuk, C. Klein

TL;DR

The paper addresses the two-dimensional Serre-Green-Naghdi equations by developing a Fourier spectral solver with GMRES to handle the elliptic step and RK4 time integration for generalized potential flows with vanishing curl. It demonstrates through extensive numerical experiments that line solitary waves are transversely stable, while no stable two-dimensional localized structures arise, resembling defocusing behavior akin to KP II dynamics. Localized initial data tend to disperse into expanding annular patterns, and radially symmetric centers can be described by an exact radially symmetric SGN solution, with no cavitation observed. These findings advance understanding of 2D SGN dynamics and provide a robust computational framework for exploring dispersive shallow-water phenomena.

Abstract

A detailed numerical study of solutions to the Serre-Green-Naghdi (SGN) equations in 2D with vanishing curl of the velocity field is presented. The transverse stability of line solitary waves, 1D solitary waves being exact solutions of the 2D equations independent of the second variable, is established numerically. The study of localized initial data as well as crossing 1D solitary waves does not give an indication of existence of stable structures in SGN solutions localized in two spatial dimensions. For the numerical experiments, an approach based on a Fourier spectral method with a Krylov subspace technique is applied.

Numerical study of the Serre-Green-Naghdi equations in 2D

TL;DR

The paper addresses the two-dimensional Serre-Green-Naghdi equations by developing a Fourier spectral solver with GMRES to handle the elliptic step and RK4 time integration for generalized potential flows with vanishing curl. It demonstrates through extensive numerical experiments that line solitary waves are transversely stable, while no stable two-dimensional localized structures arise, resembling defocusing behavior akin to KP II dynamics. Localized initial data tend to disperse into expanding annular patterns, and radially symmetric centers can be described by an exact radially symmetric SGN solution, with no cavitation observed. These findings advance understanding of 2D SGN dynamics and provide a robust computational framework for exploring dispersive shallow-water phenomena.

Abstract

A detailed numerical study of solutions to the Serre-Green-Naghdi (SGN) equations in 2D with vanishing curl of the velocity field is presented. The transverse stability of line solitary waves, 1D solitary waves being exact solutions of the 2D equations independent of the second variable, is established numerically. The study of localized initial data as well as crossing 1D solitary waves does not give an indication of existence of stable structures in SGN solutions localized in two spatial dimensions. For the numerical experiments, an approach based on a Fourier spectral method with a Krylov subspace technique is applied.
Paper Structure (15 sections, 23 equations, 17 figures)

This paper contains 15 sections, 23 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Basic facts
  3. Introduction of the potential $\varphi$
  4. Governing equations of potential flows of the SGN equations
  5. Conserved quantities
  6. 1D solitary waves
  7. Numerical approach
  8. Transverse stability of line solitary waves
  9. Deformed line solitary wave
  10. Line solitary wave plus Gaussian perturbation
  11. Initial data of crossing solitary waves
  12. Localized initial data
  13. Gaussian initial data
  14. Radially symmetric solutions
  15. Conclusion

Figures (17)

  • Figure 1: Propagation of the line soliton with $c=1.7$. We show on the left the relative conservation of the numerically computed energy, and on the right the difference between exact and numerical solution for $h$.
  • Figure 2: Solution $h$ to the 2D SGN equation for initial data being a deformed line solitary wave of the form (\ref{['soldef']}) for several values of time.
  • Figure 3: Solution $u_{x}$ to the 2D SGN equation for initial data being a deformed line solitary wave of the form (\ref{['soldef']}) for several values of time.
  • Figure 4: Solution $u_{y}$ to the 2D SGN equation for initial data being a deformed line solitary wave of the form (\ref{['soldef']}) for several values of time.
  • Figure 5: Difference between the solution to the 2D SGN equation for initial data being a deformed line solitary wave of the form (\ref{['soldef']}) for $t=10$ and an unperturbed line solitary wave centered at $x_{s}=6.995$: on the left for $h$, on the right for $u_{x}$.
  • ...and 12 more figures