An efficient model for periodic overturning waves

Numerical models for unsteady highly nonlinear water waves are presented. Two methods of approximation, linear and cubic elements, are used to discretize the Cauchy integral equations and the stability of the discretized system is shown to be dramatically enhanced by imposing feasible conditions. Fourth order accurate Lagrangian time derivatives are employed to obtain new positions of particles and the updated velocity potentials for the corresponding particles. After testing the models on the steady progressive waves, they are found to be efficient and accurate. Overturning waves on both infinite and finite depths are also presented.