Fast accurate computation of the fully nonlinear solitary surface gravity waves

Didier Clamond (1) , Denys Dutykh (2, 3, 4, 5)

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

In this short note, we present an easy to implement and fast algorithm for the computation of the steady solitary gravity wave solution of the free surface Euler equations in irrotational motion. First, the problem is reformulated in a fixed domain using the conformal mapping technique. Second, the problem is reduced to a single equation for the free surface. Third, this equation is solved using Petviashvili's iterations together with pseudo-spectral discretisation. This method has a super-linear complexity, since the most demanding operations can be performed using a FFT algorithm. Moreover, when this algorithm is combined with the multi-precision arithmetics, the results can be obtained to any arbitrary accuracy.

Mots clés: Solitary gravity waves; Fully nonlinear water wave equations; Petviashvili method

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Physique Numérique [physics.comp-ph]; Dynamique des Fluides [physics.flu-dyn]

Didier Clamond, Denys Dutykh. Fast accurate computation of the fully nonlinear solitary surface gravity waves. Computers and Fluids, 2013, 84, pp.35-38. ⟨10.1016/j.compfluid.2013.05.010⟩. ⟨hal-00759812v2⟩