  We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for
an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology.
SRF was recently introduced as an unstructured mesh formulation of Hamilton's
Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry.
In this paper, we apply the SRF equations to a representative double-lobed
axisymmetric piecewise flat geometry with mirror symmetry at the neck similar
to the geometry studied by Angenent and Knopf (A-K). We choose a specific
radial profile and compare the SRF equations with the corresponding
finite-difference solution of the continuum A-K RF equations. The
piecewise-flat 3-geometries considered here are built of
isosceles-triangle-based frustum blocks. The axial symmetry of this model
allows us to use frustum blocks instead of tetrahedra. The 2-sphere
cross-sectional geometries in our model are regular icosahedra. We demonstrate
that, under a suitably-pinched initial geometry, the SRF equations for this
relatively low-resolution discrete geometry yield the canonical Type-1 neck
pinch singularity found in the corresponding continuum solution. We adaptively
remesh during the evolution to keep the circumcentric dual lattice
well-centered. Without such remeshing, we cannot evolve the discrete geometry
to neck pinch. We conclude with a discussion of future generalizations and
tests of this SRF model.
