Efficient linear schemes for the nonlocal Cahn–Hilliard equation of phase field models

Abstract

In this paper, we develop two second-order in time, linear and unconditionally energy stable time marching schemes for solving the nonlocal Cahn–Hilliard phase field model. The main challenge to construct efficient and unconditional energy stable schemes for this model is how to design proper temporal discretizations for the nonlocal term that is induced from a convolutional type potential and the nonlinear cubic term that is induced from the double-well bulk potential. We solve these issues by developing two efficient time-stepping schemes using the Invariant Energy Quadratization approach. Its novelty is that all nonlinear terms can be treated semi-explicitly to produce linear schemes. We further show the well-posedness of the resulting linear system as well as its unconditional energy stability rigorously. Various numerical simulations are presented to demonstrate the stability, accuracy, and efficiency of the proposed schemes.