A simple, fast and stabilized flowing finite volume method for solving general curve evolution equations

TL;DR

This paper introduces a simple, fast, and stable flowing finite volume method for solving general plane curve evolution equations, combining high-order terms, anisotropy, and external fields with efficient semi-implicit discretization.

Contribution

It presents a novel Lagrangian finite volume approach with stabilization and efficient linear system solutions for complex curve evolution equations.

Findings

Method effectively handles nonlinear, anisotropic, and regularized flows.

Numerical experiments demonstrate stability and accuracy.

Applicable to a wide range of geometric evolution problems.

Abstract

A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semi-implicit in time discretization it can be numerically approximated by a solution to linear penta-diagonal systems of equations (in presence of the fourth order terms) or tri-diagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed.