A Line Element Algorithm for Curve Flow Problems in the Plane

In this paper we will describe a numerical method for the solution of curve ow problems in which the normal velocity of the curve depends locally on the position, normal and curvature of the curve. The method involves approximating the curve by a number of line elements (segments) which are only allowed to move in a direction normal to the element. Hence the normal of each line element remains constant throughout the evolution. In regions of high curvature elements naturally tend to accumulate. The method easily deals with the formation of cusps as found in ame propagation problems and is computationally comparable to a naive marker particle method. As a test of the method we present a number of numerical experiments related to mean curvature ow and ows associated with ame propagation and bushhres. 1. Introduction Suppose that at time t we have a region t in the plane with boundary @ t. We are interested in the evolution in time of the region t such that the velocity of the boundary @ t , at any point, is normal to the boundary and has a speed depending on its position in R 2 , on the direction of the normal and on the curvature of the boundary at that point. Speciically, if we can parameterize the boundary so that