On Efficient Second Order Stabilized Semi-Implicit Schemes for the Cahn-Hilliard Phase-Field Equation

TL;DR

This paper introduces two second-order semi-implicit schemes for the Cahn-Hilliard equation that are energy stable, efficient, and accurate, with rigorous analysis and numerical validation demonstrating their effectiveness.

Contribution

The paper develops and analyzes two linearly stabilized second-order schemes for the Cahn-Hilliard equation, ensuring energy stability and efficiency with proven error bounds.

Findings

Both schemes are energy stable and second-order accurate in time.

Numerical results confirm the schemes' efficiency and accuracy.

Error analysis shows second-order convergence with small time steps.

Abstract

Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank-Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of $1/\varepsilon$, where $\varepsilon$ is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed schemes.