Computation of large amplitude water waves

In this proposal we analyze research topics related to the water waves problem which is described by the incompressible Euler equations with a free boundary. We focus on a new analytical approach and a computational algorithm, based on the penalization method, for computing large amplitude water waves. This analysis has been, also, triggered by the research on wave-current interactions. In particular, we will develop an iterative algorithm in three research directions, each one supported from the relevant analytical results: (a) We will provide an iterative algorithm, which relies on second and third order asymptotic expansions, to obtain waves of large amplitude. (b) We will incorporate the effect that the variation of the total mechanical energy of the wave has on the analytical results. This will provide more robustness to the algorithm. (c) We will use the core idea of the analytical approach to restructure our algorithm in order to obtain waves of maximal amplitude. This is the first deterministic algorithm applied to this type of fluid problems which moreover is globally convergent. This approach, also, yields solutions for families of water wave problems with a big range of different properties. Among these problems the one that exhibits particular interest is related with the varying vorticity of the flow. Finally, through this algorithm we can readily obtain important characteristics of the water flows, such as the velocity field and the pressure beneath the wave, which are necessary for experimental simulations.

In this project we analyse research topics related to the water waves problem which is described by the incompressible Euler equations with a free boundary. In particular, we focus on new analytical methods and computational algorithms for calculating large amplitude water waves. Our work is pivoting around the fact that the special properties of the analytical approximations are inherited to the computational solution of the problem.