The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. We demonstrate our algorithms by computing the solution to a variety of surface motion problems. In many physical problems, interfaces move with a speed that depends on the local curvature.
Some common examples are flame propagation, crystal growth, and oil-water boundaries.
We idealize the front as a closed, non-intersecting, initial hypersurface flowing along its gradient field with a speed that depends on the curvature. Because explicit solutions seldom exist, numerical approximations are often used. In this paper, we review some recent work on algorithms for attacking these problems. We show that algorithms based on direct parameterizations of the moving front face considerable difficulties. This is because such algorithms adhere to local properties of the solution, rather than the global structure.