Energy Stable Second Order Linear Schemes for the Allen-Cahn Phase-Field Equation

TL;DR

This paper introduces two energy-stable, second-order linear schemes for the Allen-Cahn phase-field equation, improving numerical stability and efficiency in simulating interface dynamics in fluid and material sciences.

Contribution

It proposes novel stabilized semi-implicit linear schemes that are unconditionally energy stable and achieve optimal second-order convergence in time.

Findings

Schemes are unconditionally energy stable.

Numerical results verify second-order accuracy.

Applicable to 2D and 3D problems.

Abstract

Phase-field model is a powerful mathematical tool to study the dynamics of interface and morphology changes in fluid mechanics and material sciences. However, numerically solving a phase field model for a real problem is a challenge task due to the non-convexity of the bulk energy and the small interface thickness parameter in the equation. In this paper, we propose two stabilized second order semi-implicit linear schemes for the Allen-Cahn phase-field equation based on backward differentiation formula and Crank-Nicolson method, respectively. In both schemes, the nonlinear bulk force is treated explicitly with two second-order stabilization terms, which make the schemes unconditional energy stable and numerically efficient. By using a known result of the spectrum estimate of the linearized Allen-Cahn operator and some regularity estimate of the exact solution, we obtain an optimal second order convergence in time with a prefactor depending on the inverse of the characteristic interface thickness only in some lower polynomial order. Both 2-dimensional and 3-dimensional numerical results are presented to verify the accuracy and efficiency of proposed schemes.