Numerical methods for the solution of free interface problems are reviewed. For two-dimensional problems, an application of the random vortex method is proposed in which the rotational and irrotational flows are first calculated and then reconstituted into the time-dependent velocity field through the use of Hodge's decomposition theorem. The irrotational part is calculated by conformally mapping the flow, bounded on one side by the interface, into a strip at every time step, followed by use of the Gram-Schmidt orthonormalization process to solve Laplace's equation for the velocity potential. An alternative for the irrotational flow calculation, in which the free interface is represented by a vortex sheet and the boundary integral method is applied, is also discussed. The rotational field is calculated by generating vortex sheets to satisfy the no-slip boundary conditions, and by following the convective and diffusive motion of the sheets and vortex blobs. The technique is shown to yield accurate results for damping of solitary waves on shallow liquids.
