Structure-preserving numerical methods for constrained gradient flows of planar closed curves with explicit tangential velocities

TL;DR

This paper develops structure-preserving numerical methods for constrained gradient flows of planar closed curves, ensuring energy dissipation and constraint preservation using discrete gradients, Galerkin discretization, and stabilization techniques.

Contribution

The paper introduces novel structure-preserving schemes that maintain energy dissipation and constraints for gradient flows of curves, incorporating discrete gradients, Galerkin methods, and stabilization.

Findings

The proposed schemes effectively preserve energy structures.

Numerical examples demonstrate good control point distribution.

Methods achieve stability and constraint preservation.

Abstract

In this paper, we consider numerical approximation of constrained gradient flows of planar closed curves, including the Willmore and the Helfrich flows. These equations have energy dissipation and the latter has conservation properties due to the constraints. We will develop structure-preserving methods for these equations that preserve both the dissipation and the constraints. To preserve the energy structures, we introduce the discrete version of gradients according to the discrete gradient method and determine the Lagrange multipliers appropriately. We directly address higher order derivatives by using the Galerkin method with B-spline curves to discretize curves. Moreover, we will consider stabilization of the schemes by adding tangential velocities. We introduce a new Lagrange multiplier to obtain both the energy structures and the stability. Several numerical examples are presented to verify that the proposed schemes preserve the energy structures with good distribution of control points.