Cahn-Hilliard on Surfaces: A Numerical Study

TL;DR

This paper investigates a numerical scheme for the Cahn-Hilliard system on surfaces, demonstrating that high-order time discretization ensures convergence primarily dependent on spatial resolution.

Contribution

It introduces a closest-point based numerical scheme for Cahn-Hilliard on surfaces and analyzes its convergence properties with high-order time discretization.

Findings

Convergence depends mainly on spatial resolution with high-order time discretization.

The scheme effectively solves the coupled second-order systems instead of the original fourth-order.

Preconditioning with incomplete Schur-decomposition improves computational efficiency.

Abstract

The Cahn-Hilliard system has been used to describe a wide number of phase separation processes, from co-polymer systems to lipid membranes. In this work the convergence properties of a closest-point based scheme is investigated. In place of solving the original fourth-order system directly, two coupled second-order systems are solved. The system is solved using an incomplete Schur-decomposition as a preconditioner. The results indicate that with a sufficiently high-order time discretization the method only depends on the underlying spatial resolution.