Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity

TL;DR

This paper studies the evolution of plane curves driven by curvature-dependent velocities, analyzing PDE systems, proving existence and uniqueness, and presenting stable numerical schemes with computational examples.

Contribution

It introduces a class of curvature adjusted tangential velocities for geometric flows, proving theoretical properties and developing numerical methods with practical examples.

Findings

Proved local existence and uniqueness of solutions.

Developed a stable finite volume numerical scheme.

Presented computational results for nonlocal geometric flows.

Abstract

In this paper we investigate a time dependent family of plane closed Jordan curves evolving in the normal direction with a velocity which is assumed to be a function of the curvature, tangential angle and position vector of a curve. We follow the direct approach and analyze the system of governing PDEs for relevant geometric quantities. We focus on a class of the so-called curvature adjusted tangential velocities for computation of the curvature driven flow of plane closed curves. Such a curvature adjusted tangential velocity depends on the modulus of the curvature and its curve average. Using the theory of abstract parabolic equations we prove local existence, uniqueness and continuation of classical solutions to the system of governing equations. We furthermore analyze geometric flows for which normal velocity may depend on global curve quantities like the length, enclosed area or total elastic energy of a curve. We also propose a stable numerical approximation scheme based on the flowing finite volume method. Several computational examples of various nonlocal geometric flows are also presented in this paper.