A Penalization Method for Calculating the Flow Beneath Traveling Water Waves of Large Amplitude

We suggest an algorithm for the calculation of large-amplitude traveling water waves in irrotational flows and in flows with constant and nonconstant vorticity, respectively. The algorithm is based on a quite general partial differential equation constrained optimization formulation, maximizing the wave amplitude subject to the PDE constraint that the Euler equations hold. Numerical examples illustrate the performance of the algorithm for irrotational flows and in flows with constant vorticity, where the results are compared with the literature. As a proof of principle we also consider a test case with a nonconstant vorticity.