Numerical Study of the Cahn-Hilliard Equation in One, Two and Three Dimensions

TL;DR

This paper introduces a stable, efficient finite difference scheme for solving the Cahn-Hilliard equation across one, two, and three dimensions, addressing previous computational challenges and enabling larger time steps.

Contribution

A novel conservative finite difference scheme with splitting potential and free boundary conditions for the Cahn-Hilliard equation, applicable in multiple dimensions.

Findings

Achieves gradient stability in 1D, 2D, and 3D

Allows larger time steps than traditional methods

Demonstrates effectiveness across multiple dimensions

Abstract

The Cahn-Hilliard equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff an the difficulty to solve it numerically increases with the dimensionality and therefore, there are several published numerical studies in one dimension (1D), dealing with different approaches, and much fewer in two dimensions (2D). In three dimensions (3D) there are very few publications, usually concentrate in some specific result without the details of the used numerical scheme. We present here a stable and fast conservative finite difference scheme to solve the Cahn-Hilliard with two improvements: a splitting potential into a implicit and explicit in time part and a the use of free boundary conditions. We show that gradient stability is achieved in one, two and three dimensions with large time marching steps than normal methods.