  We analyze the continuum limit of a thresholding algorithm for motion by mean
curvature of one dimensional interfaces in various space-time discrete regimes.
The algorithm can be viewed as a time-splitting scheme for the Allen-Cahn
equation which is a typical model for the motion of materials phase boundaries.
Our results extend the existing statements which are applicable mostly in
semi-discrete (continuous in space and discrete in time) settings. The
motivations of this work are twofolds: to investigate the interaction between
multiple small parameters in nonlinear singularly perturbed problems, and to
understand the anisotropy in curvature for interfaces in spatially discrete
environments. In the current work, the small parameters are the the spatial and
temporal discretization step sizes $\triangle x = h$ and $\triangle t = \tau$.
We have identified the limiting description of the interfacial velocity in 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 case produces no motion (complete pinning).
