Application of nonlinear iteration scheme to the nonlinear water wave problem: Stokian wave

A new mathematical formulation for the nonlinear wave profile based on the Banach fixed point theorem is proposed. The mathematical convergence of the proposed nonlinear iteration scheme is proven in this study. As for its first application, we carried out the numerical study with Stokes' first-order solution and succeeded in solving the nonlinear wave profile of a high-order Stokian wave, that is, using only Stokes' first-order velocity potential, it is possible to realize the corresponding high-order Stokian wave profile with the help of the proposed method of iteration. Furthermore, the nonlinear strategy of iteration has a very fast convergent rate: only about 6-10 iterations are required to obtain a numerically converged solution.