A comparative study of bi-directional Whitham systems

Abstract

In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties. In the current work, we review some of the existing fully dispersive systems. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

Keywords

Bibliographic record

BibTeX Citation

@article{Dinvay2019comparativestudy,
  author = {Dinvay, E. and Dutykh, D. and Kalisch, H.},
  title = {A comparative study of bi-directional Whitham systems},
  journal = {Appl. Numer. Math.},
  year = {2019},
  volume = {141},
  pages = {248--262},
  doi = {10.1016/j.apnum.2018.09.016},
  abstract = {In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties. In the current work, we review some of the existing fully dispersive systems. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.},
  keywords = {Whitham equation, bi-directional wave propagation, surface water-wave modeling, Euler equations, weakly nonlinear approximation, two-way wave systems}
}