Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation

TL;DR

This paper introduces a new linear Crank-Nicolson scheme for the Cahn-Hilliard equation that is unconditionally energy stable, second-order accurate, and computationally efficient, with rigorous error analysis and numerical validation.

Contribution

It proposes a stabilized linear scheme with explicit nonlinear terms and proves its unconditional stability and second-order accuracy for small time steps.

Findings

Unconditionally energy stable scheme for Cahn-Hilliard equation

Second-order accuracy in time with small step sizes

Numerical results confirm efficiency and accuracy

Abstract

Efficient and unconditionally stable high order time marching schemes are very important but not easy to construct for nonlinear phase dynamics. In this paper, we propose and analysis an efficient stabilized linear Crank-Nicolson scheme for the Cahn-Hilliard equation with provable unconditional stability. In this scheme the nonlinear bulk force are treated explicitly with two second-order linear stabilization terms. The semi-discretized equation is a linear elliptic system with constant coefficients, thus robust and efficient solution procedures are guaranteed. Rigorous error analysis show that, when the time step-size is small enough, the scheme is second order accurate in time with aprefactor controlled by some lower degree polynomial of $1/\varepsilon$. Here $\varepsilon$ is the interface thickness parameter. Numerical results are presented to verify the accuracy and efficiency of the scheme.