HYPERSURFACES MOVING WITH CURVATURE-DEPENDENT SPEED: HAMILTON-JACOBI EQUATIONS, CONSERVATION LAWS AND NUMERICAL ALGORITHMS

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 model 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. The goal of this paper is to 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. Conversely, the global properties of the motion can be captured by imbedding the surface in a higher-dimensi onal function. In this setting, the equations of motion can be solved using numerical techniques borrowed from hyperbolic conservation laws. We use these schemes to follow a variety of propagation problems, illustrating corner formation, breaking and merging.