Variational approaches to water wave simulations

Abstract

This thesis starts with the study the theoretical aspects of water wave modelling using a variational framework, which is directly associated with phase space and energy conservation laws. In particular, we focus on a new variational model based on the work of Cotter and Bokhove. The new model includes accurate dispersion and incorporates the vertical component of vorticity and breaking waves modelled as bores. The Hamiltonian structure of the new water wave model is derived. A novel approach to find jump conditions at a bore is presented. In Chapter 3 we use a variational approach to derive a class of symplectic time integrators. We start with a Hamiltonian system with coordinate and momentum as canonically conjugated variables. A discrete variational principle in time is constructed, which is discretized with a discontinuous Galerkin finite element method. Following this approach we are able to obtain known time integrators as well as some new integrators. The next topic considered in this thesis is the construction of a numerical wave tank, which is discussed in Chapter 4. A second order accurate variational finite element method for the nonlinear potential flow water wave equations is developed. An important aspect is that the free surface and wave maker motion changes the computational domain each time step. A detailed validation using experimental wave tank data demonstrates the excellent capabilities of the numerical method to simulate realistic wave experiments. In particular, wave focussing is simulated accurately, catching both the location and time when a freak wave occurred in the experiment. The results for an uneven bottom and irregular waves are also found to be excellent. These results provide a suitable basis for the extension of the method to a fast moving ship in waves. In Chapter 4 we also combine the novel third order time integrator developed in Chapter 3 with the discrete variational method constructed earlier in Chapter 4 for nonlinear free surface water waves. A long time calculation reveals no energy or amplitude decay, which provides an adequate base for a further usage of the third order time integrator for more complicated water wave problems.