1: Introduction --; 1.1 Nature and scope of the work --; 1.2 Methodology --; 1.3 Innovations and conclusions --; 2: General aspects of incompressible flow. Theoretical review --; 2.1 Introduction --; 2.2 The Navier-Stokes equations for uniform, incompressible fluids --; 2.3 Initial and boundary conditions --; 2.4 The energy equation --; 2.5 The vorticity equation --; 2.6 The pressure Poisson equation for incompressible flows --; 2.7 General aspects of turbulent flows. Averaging methods and Reynolds equations --; 2.8 Turbulence transport equations --; 2.9 Turbulence models --; 2.10 Boundary conditions for K and? --; 3: Mathematical modeling of breaking shallow water waves. Proposed methodology --; 3.1 Introduction --; 3.2 Physical processes --; 3.3 Mathematical descriptions --; 3.4 Wave theories for very shallow water --; 3.5 Summary of experimental investigations --; 3.6 Description of the proposed methodology --; 4: MAC-type methods for incompressible free-surface flows --; 4.1 Introduction --; 4.2 The choice of the mesh --; 4.3 The MAC (Marker-And-Cell) method --; 4.4 The projection method --; 4.5 The SMAC (Simplified-Marker-And-Cell) method --; 4.6 The pressure-velocity iteration method --; 4.7 Numerical treatment of free-surfaces --; 4.8 Stability considerations --; 4.9 Conclusions --; 5: Description of the numerical model --; 5.1 Introduction --; 5.2 Momentum equation approximations --; 5.3 Continuity equation approximation --; 5.4 Approximations for the K and? equations --; 5.5 Updating the fluid configuration --; 5.6 Velocity boundary conditions --; 5.7 Boundary conditions for the K and? equations --; 5.8 Initial conditions for the K and? equations --; 5.9 Stability considerations --; 5.10 Programming considerations --; 5.11 Selected test problems --; 6: Numerical simulation of shallow water waves --; 6.1 Introduction --; 6.2 Propagation of a solitary wave over a horizontal bottom --; 6.3 Collision between solitary waves --; 6.4 Simulation of undular, transitional and turbulent hydraulic jumps --; 6.5 Breaking of a solitary wave over a slope --; 6.6 Breaking of a train of solitary waves over a slope --; 7: Conclusions. Future research and development --; 7.1 Summary and conclusions --; 7.2 Future research and development --; References.