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We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) foran axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology.SRF was recently introduced as an unstructured mesh formulation of Hamilton'sRicci flow (RF). It describes the RF of a piecewise-flat simplicial geometry.In this paper, we apply the SRF equations to a representative double-lobedaxisymmetric piecewise flat geometry with mirror symmetry at the neck similarto the geometry studied by Angenent and Knopf (A-K). We choose a specificradial profile and compare the SRF equations with the correspondingfinite-difference solution of the continuum A-K RF equations. Thepiecewise-flat 3-geometries considered here are built ofisosceles-triangle-based frustum blocks. The axial symmetry of this modelallows us to use frustum blocks instead of tetrahedra. The 2-spherecross-sectional geometries in our model are regular icosahedra. We demonstratethat, under a suitably-pinched initial geometry, the SRF equations for thisrelatively low-resolution discrete geometry yield the canonical Type-1 neckpinch singularity found in the corresponding continuum solution. We adaptivelyremesh during the evolution to keep the circumcentric dual latticewell-centered. Without such remeshing, we cannot evolve the discrete geometryto neck pinch. We conclude with a discussion of future generalizations andtests of this SRF model.

Simplicial Ricci Flow: An Example of a Neck Pinch Singularity in 3D