Stable Methods for Vortex Sheet Motion in the Presence of Surface Tension

Abstract

Boundary integral techniques provide a convenient way to study the evolution of an interface between inviscid liquids. Several studies have revealed that standard numerical approximations tend to lead to unstable methods, and various remedies have been introduced and tested. In this paper, we conduct a stability analysis of the linearized equations with a particular objective in mind---the determination of how the discrete system fails to capture the physical dispersion relation precisely for the available discrete modes. We discover two reasons for the typical failure in numerical discretizations: one is the inability of the mesh to represent the vorticity created by surface tension effects on the finest scale; and the other is the inaccuracies in the evaluation of the boundary integral for the velocity. With the insight gained from our linear analysis, we propose a new method that is spectrally accurate and linearly stable. Further, the exact dispersion relation is obtained for all the available discrete modes. Numerical tests suggest that the method is also stable in the nonlinear regime. However, our method runs into difficulties generic to methods based on Lagrangian motion. The markers accumulate near a stagnation point on the interface, forcing us to use an ever decreasing time step in our explicit method. We introduce a redistribution of markers to overcome this difficulty. When we redistribute according to equal arclength, we find excellent agreement with a method based on preserving equal spacing in arclength.