  We present a numerical study of the local stability of mean curvature flow of
rotationally symmetric, complete noncompact hypersurfaces with Type-II
curvature blowup. Our numerical analysis employs a novel overlap method that
constructs "numerically global" (i.e., with spatial domain arbitrarily large
but finite) flow solutions with initial data covering analytically distinct
regions. Our numerical results show that for certain prescribed families of
perturbations, there are two classes of initial data that lead to distinct
behaviors under mean curvature flow. Firstly, there is a "near" class of
initial data which lead to the same singular behaviour as an unperturbed
solution; in particular, the curvature at the tip of the hypersurface blows up
at a Type-II rate no slower than $(T-t)^{-1}$. Secondly, there is a "far" class
of initial data which lead to solutions developing a local Type-I nondegenerate
neckpinch under mean curvature flow. These numerical findings further suggest
the existence of a "critical" class of initial data which conjecturally lead to
mean curvature flow of noncompact hypersurfaces forming local Type-II
degenerate neckpinches with the highest curvature blowup rate strictly slower
than $(T-t)^{-1}$.
