Wave-type threshold dynamics and the hyperbolic mean curvature flow

Abstract

We introduce a method for computing interfacial motions governed by curvature dependent acceleration. Our method is a thresholding algorithm of the BMO-type which, instead of utilizing a diffusion process, thresholds evolution by the wave equation to obtain the desired interfacial dynamics. We also develop the numerical method and present results of its application, including an investigation of the volume preserving and multiphase motions.

On the inclusion of damping terms in the hyperbolic MBO algorithm

“…The original threshold dynamical (TD) algorithm (the so-called MBO algorithm, see [5]) is a method for approximating motion by mean curvature flow (MCF). Borrowing on such ideas, a TD algorithm for hyperbolic mean curvature flow was introduced in [3]. Whereas previous TD algorithms utilize properties of the diffusion equation to approximate MCF, properties of wave propagation (along with a particular choice of initial condition) were used to design an approximation method for HMCF.…”

The Cauchy problem of dissipative hyperbolic mean curvature flow

“…Wang [21] simplified the hyperbolic affine invariant curvature flow with dissipation term into a nonlinear strictly hyperbolic equation by using graph and support equation, respectively. The research on hyperbolic curvature flow and its related fields in earlier studies [22][23][24][26][27][28] are of great significance for many disciplines such as modern physics, films and foams, cells growing new tissue, and imaging science.…”

Discrete hyperbolic curvature flow in the plane

“…As regards the numerical approximation of hyperbolic geometric evolution equations in the literature, we are only aware of the works [20] and [9]. In the former an algorithm for the evolution of polygonal curves under crystalline hyperbolic curvature flow is presented, which corresponds to (1.1) for a crystalline, anisotropic surface energy.…”

“…In the former an algorithm for the evolution of polygonal curves under crystalline hyperbolic curvature flow is presented, which corresponds to (1.1) for a crystalline, anisotropic surface energy. On the other hand, in [9] a level-set approach, which is based on a threshold algorithm of BMO type, is used for the numerical solution of (1.5).…”