Final Report Summary - BOUSS (Theory and numerical analysis for Boussinesq systems with applications in coastal hydrodynamics) The simulation of water waves in realistic and complex environments is a very challenging problem. Most of the applications arise from the areas of coastal and naval engineering, but also from natural hazards assessment. These applications may require well-posed mathematical models in bounded domains for the computation of the wave generation, propagation, interaction with solid bodies, the computation of long-wave run-up and even the extraction of the wave energy. Issues like wave breaking, robustness of the numerical algorithm in wet-dry processes along with the validity of the mathematical models in the near-shore zone are some basic problems in this direction. During the period of the project, we studied by analytical and numerical means water-wave mathematical models and water-wave phenomena that are important in coastal hydrodynamics and applications in the tsunami generation and propagation problems. Specifically, we studied some Boussinesq systems in one and two dimensions. The standard Galerkin finite element methods (FEM) were studied for several initial-boundary value problems of Boussinesq systems including reflective boundary conditions of physical importance while in some cases the stability and convergence of the fully discrete schemes were proved. Series of experiments concerning solitary wave dynamics such as the propagation of solitary waves, the head on collision, the overtaking collision, and the effects of the boundary conditions to the long waves were performed. In the case of two dimensions, the well-possedness of some initial-boundary value problems of physical interest for Boussinesq systems of Bona-Smith type in complex plain domains (including for example obstacles) was also proved rigorously. Furthermore, thoroughly in the case of reflective boundary conditions a new modified Galerkin FEM method was constructed and studied. This numerical method can be used with any kind of structured or unstructured meshes and any kind of finite element spaces. We developed the analogous computer code and we performed several numerical experiments. Another research topic studied thoroughly was the dispersive run-up. For this study, a novel dispersive model was introduced, characterised by the very important property of the invariance under vertical translations (and therefore is more appropriate for the numerical simulation of the run-up of long waves on non-uniform shores). Afterwards, we studied thoroughly the application of some finite volume methods on non-linear dispersive wave equations while we used the novel system to study the dispersive run-up. The results were compared with experimental data and with the analogous solutions of the shallow water equations. The results showed the importance of the dispersion even in the near shore zone while new tools are provided to the community for the use of dispersive wave equations for the study of the long dispersive waves from deep waters up to the near shore region. Finally, the research was focused on the problem of tsunami wave generation. During this study a new model for the construction of dynamic co-seismic seabed displacements was proposed. This model is based on the finite fault solution for the slip distribution under some assumptions on the dynamics of the rupturing process. Once the bottom motion is reconstructed, we studied waves induced on the free surface of the ocean. For this purpose we considered three different models approximating the Euler equations of the water-wave theory. Namely, we used the linearised Euler equations, a Boussinesq system and a novel weakly nonlinear model. An intercomparison of these approaches was performed in the case of 17 July 2006 Java event, where an underwater earthquake of magnitude 7.7 generated a tsunami that inundated the southern coast of Java. All computations can be done efficiently enough and can be used in a real-time tsunami warning system.