Localization in Spatial-Spectral Method for Water Wave Applications

In the description of water waves, dispersion is one of the most important physical properties; it specifies the propagation speed as function of the wavelength. Accurate modelling of dispersion is essential to obtain high-quality wave propagation results. The relation between speed and wavelength is given by a non-algebraic relation; for finite element/difference methods this relation has to be approximated and leads to restrictions for waves that are propagated correctly. By using a spectral implementation dispersion can be dealt with exactly above flat bottom using a pseudo-differential operator so that all wavelengths can be propagated correctly. However, spectral methods are most commonly applied for problems in simple domains, while most water wave applications need complex geometries such as (harbour) walls, varying bathymetry, etc.; also breaking of waves requires a local procedure at the unknown position of breaking. This paper deals with such inhomogeneities in space; the models are formulated using Fourier integral operators and include non-trivial localization methods. The efficiency and accuracy of a so-called spatial-spectral implementation is illustrated here for a few test cases: wave run-up on a coast, wave reflection at a wall and the breaking of a focussing wave. These methods are included in HAWASSI software (Hamiltonian Wave-Ship-Structure Interaction) that has been developed over the past years.