| The Euler equations which govern the free-surface motion of an ideal fluid (the water wave problem) are notoriously difficult to solve for a number of reasons. First, they are a classical free-boundary problem where the domain shape is one of the unknowns to be found. Another difficulty is that there is no natural dissipation mechanism (the water wave problem is a Hamiltonian system so energy is conserved) so that spurious high-frequency modes are not damped. In this thesis the latter of these difficulties is addressed using a surface formulation (which addresses the former complication) supplemented with physically-motivated viscous effects recently derived by Dias, Dyachenko, and Zakharov (2008). The novelty of this approach is to derive a weakly nonlinear model from the surface formulation of Zakharov (1968) and Craig & Sulem (1993) complemented with the viscous effects mentioned above. The new model proposed in this thesis is simple to implement while being both faithful to the physics of the problem and extremely stable numerically. |
