Forecasting water waves and currents : a space-time approach

Abstract

Forecasting water waves and currents in near shore and off shore regions of the seas and oceans is essential to maintain and protect our environment and man made structures. In wave hydrodynamics, waves can be classified as shallow and deep water waves based on its water depth. The mathe- matical models of these waves are shallow water and free surface gravity water wave equations which describe the hydrodynamics of waves and cur- rents near shore and off shore regions of seas and oceans. The complexity in these models exist as moving boundaries whose position depends on the solution of the governing equations. For shallow water waves, it is the shore line boundary where the water depth falls dry and for deep water waves, it is the free surface which separates the sea or ocean from atmospheric air. It is often difficult to solve these wave equations analytically while solving them numerically in an efficient and accurate way is a challenging task because of the moving boundaries. The numerical challenges are two fold: one is to develop a numerical method which is accurate and efficient for deforming grids and the other is to design a numerical algorithm for the grid adaptation following the moving boundaries. In this thesis, we aimed at first developing space-time discontinuous Galerkin finite element schemes for shallow water and free surface gravity water wave equations which are accurate and efficient for deforming grids.