Convergence of a threshold-type algorithm using the signed distance function

Abstract

We consider a threshold-type algorithm for curvature-dependent motions of hypersurfaces. This algorithm was numerically studied by [27], [9] and [35], where they used the signed distance function. It is also regarded as a variant of the Bence–Merriman–Osher algorithm for the mean curvature flow ( [4]). In this paper we prove the convergence of our algorithm under the nonfattening condition, applying the method of [30] which is based on the notion of the generalized flow due to [3]. Then we derive the rate of convergence of our algorithm to the smooth and compact curvature-dependent motions and show its optimality to the special case of a circle evolving by its curvature. We also give a local estimate on the convergence to a regular portion of the generalized curvature-dependent motion.