2014/11/02 by Oleksandr Misiats, Misiats, Oleksandr, Nung Kwan Yip +1 · 2 citations
Mathematics · Computer Science · #Differential Equations and Numerical Methods #Stochastic processes and statistical mechanics #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.1411.0301
We analyze the continuum limit of a thresholding algorithm for motion by mean\ncurvature of one dimensional interfaces in various space-time discrete regimes.\nThe algorithm can be viewed as a time-splitting scheme for the Allen-Cahn\nequation which is a typical model for the motion of materials phase boundaries.\nOur results extend the existing statements which are applicable mostly in\nsemi-discrete (continuous in space and discrete in time) settings. The\nmotivations of this work are twofolds: to investigate the interaction between\nmultiple small parameters in nonlinear singularly perturbed problems, and to\nunderstand the anisotropy in curvature for interfaces in spatially discrete\nenvironments. In the current work, the small parameters are the the spatial and\ntemporal discretization step sizes triangle x = h and triangle t = \τ.\nWe have identified the limiting description of the interfacial velocity in the\n(i) sub-critical (h \≪ \τ), (ii) critical (h = O(\τ)), and (iii)\nsuper-critical (h \≫ \τ) regimes. The first case gives the classical\nisotropic motion by mean curvature, while the second produces intricate pinning\nand de-pinning phenomena and anisotropy in the velocity function of the\ninterface. The last case produces no motion (complete pinning).\n