Stability of piecewise flat Ricci flow

TL;DR

This paper analyzes the stability of a novel piecewise flat Ricci flow method through linear analysis and simulations, proposing adaptations to improve stability and demonstrating convergence to smooth Ricci flow solutions.

Contribution

It introduces stability improvements for the piecewise flat Ricci flow and shows convergence to smooth solutions, advancing discrete geometric flow methods.

Findings

Stability of the piecewise flat Ricci flow is enhanced through specific adaptations.

Numerical simulations confirm convergence to smooth Ricci flow solutions.

The method is applicable to a variety of manifolds.

Abstract

The stability of a recently developed piecewise flat Ricci flow is investigated, using a linear stability analysis and numerical simulations, and a class of piecewise flat approximations of smooth manifolds is adapted to avoid an inherent numerical instability. These adaptations have also been used in a related paper to show the convergence of the piecewise flat Ricci flow to known smooth Ricci flow solutions for a variety of manifolds.