| Numerical instability is a common difficulty in simulations of interfacial flows. In addition, when surface tension is included into the model, the evolution equations of the interface become stiff, that is, stable explicit time integration methods require a severe constraint in the size of the time step, and the application of implicit methods is difficult because the surface tension enters into the interface dynamics nonlinearly and nonlocally.;Hou, Lowengrub and Shelley (J. Comp. Phys. 114:312-338, 1994) design semi-implicit time integration methods for a spectral discretization in space of a new boundary integral formulation for computing the motion of an interface between two irrotational, inviscid, and incompressible fluids in two dimensions. Their methods remove the high order stiffness and perform remarkably well even in the full nonlinear regime of motion, though they apply some numerical filtering at each time step to suppress an observed aliasing instability.;In this thesis we study the nonlinear stability and convergence of a method based on the spectral discretization of Hou et. al., keeping time continuous and including de-aliasing filtering only at suitable places. We are able to prove rigorously the convergence of the method provided the solution is sufficiently smooth. The proof relies on delicate energy estimates for the errors in discrete Sobolev spaces.;We also give an alternative formulation and a scheme for water waves which is more efficient numerically than the method for the general two-density fluid interface applied to this special case. We state a convergence theorem and outline its proof. Bottom topography is also considered.;Applications to Rayleigh-Taylor instability, droplet formation in a single fluid, and breaking waves with bottom topography are given. We find that surface tension leads to an interesting phenomenon in overturning waves: the interface curvature develops a concentrated spike at a certain time and subsequent capillary waves appear on the forward side of the crest. |
