Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth

Didier Clamond (1) , Denys Dutykh (2, 3, 4)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 LAMA - Laboratoire de Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
4 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques

This paper describes an efficient algorithm for computing steady two-dimensional surface gravity wave in irrotational motion. The algorithm complexity is O(N log N), N being the number of Fourier modes. The algorithm allows the arbitrary precision computation of waves in arbitrary depth, i.e., it works efficiently for Stokes, cnoidal and solitary waves, even for quite large steepnesses. The method is based on conformal mapping, Babenko equation rewritten in a suitable way, pseudo-spectral method and Petviashvili's iterations. The efficiency of the algorithm is illustrated via some relevant numerical examples. The code is open source, so interested readers can easily check the claims, use and modify the algorithm.

Mots clés: Babenko equation; spectral methods; Periodic waves; Petviashvili scheme; Stokes wave; cnoidal waves

Domaines: Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Physique Atmosphérique et Océanique [physics.ao-ph]; Physique Numérique [physics.comp-ph]; Dynamique des Fluides [physics.flu-dyn]; Formation de Structures et Solitons [nlin.PS]

Didier Clamond, Denys Dutykh. Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth. Journal of Fluid Mechanics, 2018, 844, pp.491-518. ⟨10.1017/jfm.2018.208⟩. ⟨hal-01465813v2⟩