50 research outputs found
Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow
This paper is concerned with diffuse-interface approximations of the Willmore
flow. We first present numerical results of standard diffuse-interface models
for colliding one dimensional interfaces. In such a scenario evolutions towards
interfaces with corners can occur that do not necessarily describe the adequate
sharp-interface dynamics.
We therefore propose and investigate alternative diffuse-interface
approximations that lead to a different and more regular behavior if interfaces
collide. These dynamics are derived from approximate energies that converge to
the -lower-semicontinuous envelope of the Willmore energy, which is in
general not true for the more standard Willmore approximation
A Monotone, Second Order Accurate Scheme for Curvature Motion
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