A Monotone, Second Order Accurate Scheme for Curvature Motion

TL;DR

This paper introduces a second order accurate, unconditionally monotone numerical scheme for curve shortening flow that preserves the comparison principle and guarantees convergence to the viscosity solution.

Contribution

It presents a novel monotone, second order in time scheme based on threshold dynamics for curvature motion, ensuring convergence and preserving comparison principles.

Findings

Scheme is unconditionally monotone and second order accurate in time.

Ensures convergence to the viscosity solution of curve shortening.

Preserves the comparison principle of the exact evolution.

Abstract

We present a second order accurate in time numerical scheme for curve shortening flow in the plane that is unconditionally monotone. It is a variant of threshold dynamics, a class of algorithms in the spirit of the level set method that represent interfaces implicitly. The novelty is monotonicity: it is possible to preserve the comparison principle of the exact evolution while achieving second order in time consistency. As a consequence of monotonicity, convergence to the viscosity solution of curve shortening is ensured by existing theory.