Seminar

In this presentation we review some methods for evolution of curves and surfaces Ωt. The methods are

1. The level set method by Sethian,
2. The fast marching method of Osher and Sethian,
3. Method based on introduction of ray coordinates associated with Ωt and
4. Kinematical Conservation Laws.

We present some results and suggest comparison of these results by intensive numerical computation. Introduction of ray coordinates simplifies tracking of the successive positions of the surface. Formulation in terms of KCL has further advantages. The first two methods have been discussed intensively both theoretically and with numerical results on many practical problems. However, these methods can not reproduce appearance of a special singularity on Ωt, namely kinks across which the normal to Ωt and the velocity of Ωt have finite jumps. Further the computational efficiency of last two methods are equally good. We also highlight a case when further theoretical development is required in order to make the last two methods applicable t ocurvature driven Ωt.