Approximation and comparison for motion by mean curvature with intersection points

Abstract Consider the motion of a curve in the plane under its mean curvature. It is a very interesting problem to investigate what happens when there are intersection points on the curve at which the mean curvature is singular. In this paper, we study this issue numerically by solving the Allen-Cahn equation and the nonlocal evolution equation with Kac potential. The Allen-Cahn equation is discretized by a monotone scheme, and the nonlocal evolution equation with Kac potential is discretized by the spectral method. Several curves with intersection points under motion by mean curvature are studied. From a simple analysis and our numerical results, we find that which direction to split of the curve at the intersection point depends on the angle of the curve at the point, i.e., it splits in horizontal direction when the angle α > π/2, in vertical direction when α

[1]  J. Sethian.  Level set methods : evolving interfaces in geometry, fluid mechanics, computer vision, and materials science , 1996 .

[2]  J. Tyson,et al.  A cellular automaton model of excitable media. II: curvature, dispersion, rotating waves and meandering waves , 1990 .

[3]  B. Hess,et al.  Isotropic cellular automaton for modelling excitable media , 1990, Nature.

[4]  Motion by curvature by scaling nonlocal evolution equations , 1993 .

[5]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .

[6]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[7]  P. Souganidis,et al.  Generalized motion by mean curvature as a macroscopic limit of stochastic ising models with long range interactions and Glauber dynamics , 1995 .

[8]  Jack Xin,et al.  Diffusion-Generated Motion by Mean Curvature for Filaments , 2001, J. Nonlinear Sci..

[9]  Enza Orlandi,et al.  Glauber evolution with Kac potentials: III. Spinodal decomposition , 1996 .

[10]  Tom Ilmanen,et al.  Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature , 1993 .

[11]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[12]  Charles M. Elliott,et al.  CONVERGENCE OF NUMERICAL SOLUTIONS TO THE ALLEN-CAHN EQUATION , 1998 .

[13]  G B Ermentrout,et al.  Cellular automata approaches to biological modeling. , 1993, Journal of theoretical biology.

[14]  Markos A. Katsoulakis,et al.  Stochastic curvature flows: asymptotic derivation, level set formulation and numerical experiments , 2001 .

[15]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[16]  J. Tyson,et al.  A cellular automation model of excitable media including curvature and dispersion. , 1990, Science.

[17]  Steven J. Ruuth.  A Diffusion-Generated Approach to Multiphase Motion , 1998 .

[18]  Enza Orlandi,et al.  Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics , 1994 .

[19]  S. Osher,et al.  Motion of multiple junctions: a level set approach , 1994 .