Numerical Simulation of a Weakly Nonlinear Model for Water Waves with Viscosity

The potential flow 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. Additionally, they are strongly nonlinear (with derivatives appearing in the nonlinearity) without a natural dissipation mechanism so that spurious high–frequency modes are not damped. In this contribution we address the latter of these difficulties 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 our 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. Our new model is simple to implement while being both faithful to the physics of the problem and extremely stable numerically.