The prognostic equations are obtained from the kinematic and dynamic boundary conditions at the interface, i.e. where , and the gravity g acts downwards. The prognostic equations are expanded in Taylor series, keeping several terms, in order to obtain an efficient scheme. Accurate solution of the Laplace equation is crucial to an algoritm for computing interfacial flows, and we here apply complex theory and Cauchy's integral theorem for this purpose. For interfacial waves with finite amplitude the physical Kelvin-Helmholtz instability is inevitably included in the theoretical model. This represents a difficulty since the space discretization cannot be too fine.
Instability is removed by a regridding procedure and smoothing.
We are, however, able to document convergence of the method and accuracy in all cases considered. The mathematical model is documented in reference 1. A body, starting from rest, is moving horizontally with speed U in the upper layer, and we shall compute the resulting motion of the interface. The resulting profiles of the interface exhibit an undular depression that is developing ahead of the body, with a number of oscillations which is increasing with time, see figure 1 . Behind the body an almost horizontal elevation develops, with length which also is increasing with time.