Energy stable auxiliary variable method for Cahn--Hilliard equations

TL;DR

This paper introduces a quadratic reformulation and a new energy-stable numerical method for the Cahn-Hilliard equation with rational-like free-energy, ensuring energy dissipation and capturing anisotropic effects.

Contribution

It develops the Quadratic Conservation Elevation (QCE) method combining SAV and midpoint rule for stable discretization of complex free-energy Cahn-Hilliard equations.

Findings

The method preserves the energy dissipation law.

Numerical results confirm efficiency and accuracy.

It captures anisotropic orientation effects.

Abstract

In this paper, we propose a quadratic reformulation theory for rational-like functions. Based on this theory, we develop the Quadratic Conservation Elevation (QCE) method, which combines the Scalar Auxiliary Variable (SAV) method with the implicit midpoint rule. We apply this approach to the Cahn-Hilliard (CH) equation with rational-like free-energy terms, obtaining numerical discretizations that preserve the original energy dissipation law. We further derive the discrete dispersion relation and coarsening dynamics, confirming the efficiency and consistency of the method with the continuous counterpart. In addition, we use the proposed method to capture missing orientations for different anisotropic functions. Numerical simulations with various initial conditions illustrate phase separation and anisotropic evolution.