A mixed Eulerian-Lagrangian High-Order Spectral method for the propagation of ocean surface waves over a flat bottom

We introduce a numerical method to describe the propagation of two-dimensional nonlinear water waves over a flat bottom. The free surface is described in terms of a Lagrangian representation, i.e. by following the position and the velocity potential of a set of surface particles. The method consists in a mixed Eulerian-Lagrangian modification of the classical High-Order Spectral (HOS) method. At each time step, the Eulerian velocity potential inside the domain and the velocity of the surface particles are estimated by using a spectral decomposition along with a perturbation expansion at an arbitrary order M. The Lagrangian description of the surface makes it possible to use lower approximation orders and fewer Fourier modes to capture steep nonlinear waves, which also improves the numerical stability of the method. Its accuracy is established for steep regular waves by comparing simulations to existing Lagrangian and Eulerian solutions, as well as to traditional HOS-simulations. For irregular bichromatic waves, we show with an example that the obtained solution converges with respect to the Lagrangian conservation equations as the order M increases. Finally, the ability of the proposed method to compute the velocity field in steep irregular waves is demonstrated.