École Nationale Supérieure de
Techniques Avancées

Abstract

The numerical simulation of wave propagation approaching the coastline is an important issue for coastal and maritime actors. Several mathematical models, with various levels of complexity, allow this to be done. Here, we use a fully nonlinear and dispersive model in a potential flow framework, based on Zakharov’s equations. In order to march these equations in time, an elliptic problem must be solved at each time step. The originality of the approach presented in this report lies in the combination of a spectral method for the vertical direction with a finite element approach for the horizontal plane. The weak formulation of the equations is implemented in FreeFEM and then validated on two test cases : one from linear theory and one from a nonlinear stationary wave. P1, P2 and P3 finite elements are studied, and the approach is improved through the use of L2 projection, which allows for a more accurate calculation of the derivatives that compose the coefficients of the equation. The handling of Neumann boundary conditions is also discussed. The results appear promising on these two cases, and need to be further confirmed on more realistic applications (coastal domains of arbitrary shape, cases with variable water depth, other types of boundary conditions, etc.).