Numerical investigation of Stokes waves: from high accuracy solutions to stability.

Abstract:  We will consider one of the first finite amplitude, meaning inherently nonlinear, solutions for waves on the surface of the fluid, namely, Stokes waves. During almost two centuries, since the first work by  Sir George Stokes in 1847, researchers were studying this fascinating solution of 2D Euler equations. Finding the limiting Stokes wave was the first application of conformal mappings to hydrodynamics, it was shown that there is a wave of highest amplitude with a 120 degrees angle on the crest. We shall first consider how to obtain these solutions numerically using conformal mappings and Newton-Conjugate Gradient method, then we shall study their analytic structure through Pade approximation via application of Alpert-Greengard-Hagstrom algorithm. And finally, we shall investigate stability of the Stokes waves using shifted Arnoldi algorithm on inhomogeneous grid.