Convergence of space-time discrete threshold dynamics toanisotropic motion by mean curvature
Oleksandr Misiats, Nung Kwan Yip · Discrete and Continuous Dynamical Systems · 2016
We analyze the continuum limit of a thresholding algorithm for motion by meancurvature of one dimensional interfaces in various space-time discreteregimes. The algorithm can be viewed as a time-splitting scheme for theAllen-Cahn equation which is a typical model for the motion of materials phaseboundaries. Our results extend the existing statements which are applicablemostly in semi-discrete (continuous in space and discrete in time) settings.The motivations of this work are twofolds: to investigate the interactionbetween multiple small parameters in nonlinear singularly perturbed problems,and to understand the anisotropy in curvature for interfaces in spatiallydiscrete environments. In the current work, the small parameters arethe spatial and temporal discretization step sizes: $\triangle x = h$ and$\triangle t = \tau$.We have identified the limiting description of the interfacial velocityin the (i) sub-critical ($h \ll \tau$),(ii) critical ($h = O(\tau)$), and(iii) super-critical ($h \gg \tau$) regimes.The first case gives the classical isotropic motion by mean curvature,while the second produces intricate pinning and de-pinning phenomena,and anisotropy in the velocity function of the interface. The last caseproduces no motion (complete pinning).