Forecasting Water Waves and Currents: A Space-time Approach

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 mathematical models of these waves are shallow water and free surface gravity water wave equations which describe the hydrodynamics of waves and currents 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. The shallow water equations are a leading order hydrodynamic model for coastal waves and currents. This is because they can exhibit the complicated flooding and drying phenomena due to the moving shore line boundary, and the wave breaking phenomena in the form of bores. A new space-time discontinuous Galerkin (DG) discretization is first presented for the (rotating) shallow water equations over varying topography and fixed boundaries. We formulated the discretization in an efficient and conservative way with the numerical HLLC flux on the finite element boundaries. We also designed a novel way to apply numerical dissipation around discontinuities, that are present in the form of bores, with the help of Krivodonova's discontinuity indicator such that the spurious oscillations are suppressed. The non-linear algebraic system resulting from the space-time discretization is solved using a pseudo-time integration method. A thorough verification of the space-time DG finite element method is undertaken by comparing the numerical and exact solutions. We carried out a discrete Fourier analysis of the one dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error. The numerical scheme is verified and validated for a number of problems arising in geophysical flows. To demonstrate that the space-time DG method is particularly suitable for problems with dynamic grid motion, we simulated nonlinear waves generated by a wave maker and verified these for low amplitude waves where linear theory is approximately valid. Free surface gravity water wave equations is widely used in marine and offshore engineering to model waves. The mathematical nature of these equations is complex because it consists of a potential flow equation which is of elliptic nature and nonlinear free surface boundary conditions which are hyperbolic in nature. Hence, a space-time discontinuous Galerkin finite element method is presented for simplified linear free surface gravity water waves. The free surface gravity water wave equations also arise from Luke's variational formulation which is associated with the conservation of energy and phase space, under suitable boundary conditions. This variational formulation also provided a basis to obtain a novel space-time variational (dis)continuous Galerkin finite element method. Both the space-time discontinuous Galerkin and the space-time variational finite element discretizations result in an algebraic linear system of equations with a very compact stencil, i.e., the algebraic equations from each element is coupled to its immediate neighboring elements only. Thus, the linear system of equations are built using an efficient block sparse matrix storage routine and solved by using iterative linear solvers using a well-tested PETSc package. Numerical schemes are verified for harmonic waves in a periodic domain and generated in a wave basin. Extension of the space-time discontinuous Galerkin method for flooding and drying in shallow water waves and nonlinear free surface evolution of deep water waves will be the topic of future research