Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature

TL;DR

This paper proves the convergence of a crystalline algorithm for simulating heat flow and curvature-driven motion of one-dimensional graphs, providing a new numerical method for quasilinear parabolic equations.

Contribution

It offers the first convergence analysis of a crystalline approximation scheme for curvature motion in one dimension, specifically for graph-like surfaces.

Findings

Convergence of the crystalline scheme is established for 1D graphs.

The scheme provides an effective numerical method for quasilinear parabolic equations.

This approach links geometric evolution laws with computational algorithms.

Abstract

Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law using a "crystalline" approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension.