Courant Institute, New York University, New York
Many fundamental problems in fluid mechanics and materials science concern the dynamics of interfaces mediated by surface tension or energy -- pattern and singularity formation are two important examples. For a computational scientist, surface tension leads to severe time-step constraints that are both nonlinear and dynamical in nature. I will discuss an approach, based on a "Small-Scale Decomposition", that for a class of such problems circumvents these difficulties. By reposing the dynamics in intrinsic coordinates, and analyzing the equations of motion at small length-scales, the contribution of the surface tension can be isolated as a linear term that can be treated implicitly, which leads to efficient and accurate time-stepping methods. These methods are illustrated through computational studies of the effect of surface tension on the Kelvin-Helmholtz instability, pattern formation problems in the Hele-Shaw cell, and others.