Discrete hyperbolic curvature flow in the plane

TL;DR

This paper introduces a finite difference method for hyperbolic curvature flow in the plane, providing error analysis and numerical simulations that reveal singularity formation from smooth convex curves.

Contribution

It develops a semidiscrete finite difference scheme for hyperbolic curvature flow and establishes error bounds, advancing numerical methods for this geometric evolution equation.

Findings

Error bounds for the numerical scheme are proven.

Numerical simulations demonstrate singularity formation.

The method accurately captures the evolution of convex curves.

Abstract

Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave phenomena in solid-liquid interfaces. We introduce a semidiscrete finite difference method for the approximation of hyperbolic curvature flow and prove error bounds for natural discrete norms. We also present numerical simulations, including the onset of singularities starting from smooth strictly convex initial data.