The Whitham equation with surface tension

E. Dinvay; D. Moldabayev; D. Dutykh; H. Kalisch

doi:10.1007/s11071-016-3299-7

2017

The Whitham equation with surface tension

NNonlinear Dynamics

E. Dinvay, D. Moldabayev, D. Dutykh and H. Kalisch

Nonlinear Dynamics 88(2): 1125-1138 (2017)

Open Access

HAL Version DOI

Abstract

The viability of the Whitham equation as a nonlocal model for capillary-gravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian structure of the free surface water wave problem and the Dirichlet-Neumann operator. The system features gravitational and capillary effects, and when restricted to one-way propagation, the system reduces to the capillary Whitham equation. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the free-surface problem for the Euler system than other models like the KdV, and Kawahara equation. In the case of relatively strong capillarity considered here, the KdV and Kawahara equations outperform the Whitham equation with surface tension only for very long waves with negative polarity.

Keywords

math.APmath.NAphysics.flu-dynphysics.comp-phphysics.ao-phMathematicsPhysicsComputer Sciencecs.NA

Bibliographic record

Journal: Nonlinear Dynamics

Volume: 88

Issue: 2

Pages: 1125-1138

DOI: 10.1007/s11071-016-3299-7

arXiv: 2002.09946

HAL: hal-01504047

BibTeX Citation

@article{Dinvay2017whithamequation,
  author = {Dinvay, E. and Moldabayev, D. and Dutykh, D. and Kalisch, H.},
  title = {The Whitham equation with surface tension},
  journal = {Nonlinear Dynamics},
  year = {2017},
  volume = {88},
  number = {2},
  pages = {1125--1138},
  doi = {10.1007/s11071-016-3299-7},
  abstract = {The viability of the Whitham equation as a nonlocal model for capillary-gravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian structure of the free surface water wave problem and the Dirichlet-Neumann operator. The system features gravitational and capillary effects, and when restricted to one-way propagation, the system reduces to the capillary Whitham equation. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the free-surface problem for the Euler system than other models like the KdV, and Kawahara equation. In the case of relatively strong capillarity considered here, the KdV and Kawahara equations outperform the Whitham equation with surface tension only for very long waves with negative polarity.},
  keywords = {math.AP, math.NA, physics.flu-dyn, physics.comp-ph, physics.ao-ph, Mathematics, Physics, Computer Science, cs.NA}
}