Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves

TL;DR

This paper introduces a semi-implicit numerical scheme for approximating elastic knots and self-avoiding inextensible curves, ensuring stability, energy decay, and preservation of arclength, with applications to finding global energy minimizers.

Contribution

It presents a novel semi-implicit scheme with proven stability for elastic knot approximation, combining bending energy and tangent-point functional, and explores energy landscapes through numerical experiments.

Findings

Energy decay during evolution

Maintenance of arclength parametrization

Insights into global minimizers of bending energy

Abstract

We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional. Based on estimates for the second derivative of the latter and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization. Finally we present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in knot classes, so-called elastic knots.