Efficient Algorithms for Diffusion-Generated Motion by Mean Curvature

The problem of simulating the motion of evolving surfaces with junctions according to some curvature-dependent speed arises in a number of applications. By alternately diffusing and sharpening characteristic functions for each region, a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums. However, the usual finite difference discretization of these methods is often excessively slow when accurate solutions are sought, even in two dimensions. We propose a new, spectral discretization of these diffusion-generated methods which obtains greatly improved efficiency over the usual finite difference approach. These efficiency gains are obtained, in part through the use of a quadrature-based refinement technique, by integrating Fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients. Indeed, numerical studies demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization. Our findings are demonstrated on several examples.