Spatial and temporal stability issues for interfacial flows with surface tension

Many physically interesting problems involve the propagation of free surfaces in fluids with surface tension effects. Surface tensions is an ever-present physical effect that is often neglected due to the difficulties associated with its inclusion in the equations of motion. Accurate simulation of these interfaces presents a problem of considerable difficulty on several levels. First, even for stably stratified flows like water waves, it turns out that straightforward spatial discretizations (of the boundary integral formulation) generate numerical instability. Second, surface tension introduces a large number of derivatives through the Laplace-Young boundary condition. This induces severe time step restrictions for explicit time integration methods. In this paper, we present a class of stable spatial discretizations and we present a reformulation of the equations of motion that make apparent how to remove the high order time step restrictions introduced by the surface tension. This paper is a review of the results given in [1,2].