A New Measure of Coarseness for Solutions to Cahn--Hilliard Equations

TL;DR

This paper introduces a novel coarseness measure for phase separation in Cahn--Hilliard equations, enabling consistent characterization over time and across different solution structures, and validates it through numerical comparisons.

Contribution

The paper presents a new coarseness measure that remains consistent during evolution and applies it to evaluate existing coarsening models against numerical data.

Findings

The new measure is effective for solutions with no periodic structure.

It provides a reliable comparison between different coarsening models.

The measure remains consistent throughout the phase separation process.

Abstract

We introduce a new measure of coarseness for characterizing phase separation processes such as those described by Cahn--Hilliard equations. An advantage of our measure is that it remains consistent throughout the evolution, including for solutions with no periodic structure. We use our measure to compare two previous models of coarsening dynamics with numerically generated dynamics, providing the first direct check that we are aware of for the efficacy of these methods.