This figure shows a numerical simulation of Lagrangian controllability. The yellow curve delimits the target zone and the blue curve is a snapshot of the edge of the zone of the portion of fluid that one moves in order to bring it onto the target. The control is done on the inner edge of the domain. One of the difficulties of simulations is to keep the differentiable character of objects. Credits: Guillaume Legendre & François-Xavier Vialard (Paris-Dauphine University)

The need to introduce an arbitrarily finite field of study to determine the propagation of waves in an infinite medium raises the problem of finding suitable transparent conditions preventing on the one hand the return of waves by bounces and on the other hand avoiding l appearance of common singularities in the study of wave propagation in stratified media in limited domain. Compared to the classical cases, the expressions of the transparent conditions have to be rewritten because they depend on the propagation conditions at the interface between media with very different physical and dynamic properties. Classic case of boundary conditions (left film) and case with new boundary conditions avoiding singularities (right film).

Bissengue, D. Conditions aux limites transparentes et modélisation des
vagues de surface dans un écoulement. Master’s thesis, EICNAM, 2012.

Orellana, J. « Non-reflecting boundary conditions for surface water wave simulation in the vicinity of a solid body » Proceedings of the fourteenth Zaragoza-Pau Conference on Mathematics and its Applications, september 2016, à paraître.

Destuynder, P., and Fabre, C. A discussion on neumann–kelvin’s model for progressive water waves. Applicable Analysis 90, 12 (2011), 1851–1876.

Use of a high-order "CLA" absorbing limit condition (Hagstrom and Warburton - 2004) dedicated to non-evanescent waves on a coupled hydrodynamic wave model (Ph. Destuynder-2010,2012) to reduce the computational domain and l ’use as an interface condition thus allowing resolution with the aid of the Schwarz wave relaxation method "SWR" (Schwarz-1870, Lelarasmee-1982, Bjørhus-1995). The difficulty due to the coupled character of the model associated with different wave speeds can be reduced by using a constraint on the CLA. The result was obtained in five iterations of "SWR". The solution of the problem (on surface "gray curve" and in the basin "in shades of gray") merges with the reference solution (on surface "black curve" and in the basin "
Wilk, Olivier - "EDP-Normandie" Caen 2013 - ISBN 978-2-9541221-2-0