Comparison study for Level set and Direct Lagrangian methods for computing Willmore flow of closed planar curves

TL;DR

This paper compares level set and direct Lagrangian methods for simulating the Willmore flow of closed planar curves, introducing new numerical schemes and analyzing their accuracy and consistency.

Contribution

It presents new semi-implicit numerical schemes for both methods and demonstrates their second-order accuracy and close agreement in various elastic curve evolutions.

Findings

Both methods are experimentally second order accurate.

The approaches coincide closely when linear systems are solved precisely.

Proper redistancing and boundary condition handling are crucial for consistency.

Abstract

The main goal of this paper is to present results of comparison study for the level set and direct Lagrangian methods for computing evolution of the Willmore flow of embedded planar curves. To perform such a study we construct new numerical approximation schemes for both Lagrangian as well as level set methods based on semi-implicit in time and finite/complementary volume in space discretizations. The Lagrangian scheme is stabilized in tangential direction by the asymptotically uniform grid point redistribution. Both methods are experimentally second order accurate. Moreover, we show precise coincidence of both approaches in case of various elastic curve evolutions provided that solving the linear systems in semi-implicit level set method is done in a precise way, redistancing is performed occasionally and the influence of boundary conditions on the level set function is eliminated.