Convergence of thresholding schemes incorporating bulk effects

TL;DR

This paper proves the convergence of three thresholding algorithms that model interface motion in multi-phase systems with bulk effects, including volume preservation, bulk energy, and boundary influences.

Contribution

It establishes the convergence of new thresholding schemes for interface motion incorporating bulk effects, extending previous algorithms like Merriman-Bence-Osher.

Findings

Proves convergence of a volume-preserving mean-curvature flow scheme.

Analyzes a scheme for surface tension plus bulk energy flow.

Models grain growth with boundary effects in polycrystals.

Abstract

In this paper we establish the convergence of three computational algorithms for interface motion in a multi-phase system, which incorporate bulk effects. The algorithms considered fall under the classification of thresholding schemes, in the spirit of the celebrated Merriman-Bence-Osher algorithm for producing an interface moving by mean curvature. The schemes considered here all incorporate either a local force coming from an energy in the bulk, or a non-local force coming from a volume constraint. We first establish the convergence of a scheme proposed by Ruuth-Wetton for approximating volume-preserving mean-curvature flow. Next we study a scheme for the geometric flow generated by surface tension plus bulk energy. Here the limit is motion by mean curvature (MMC) plus forcing term. Third we consider a thresholding scheme for simulating grain growth in a polycrystal surrounded by air, which incorporates boundary effects on the solid-vapor interface. The limiting flow is MMC on the inner grain boundaries, and volume-preserving MMC on the solid-vapor interface.