Fully-Decoupled and Second-Order Time-Accurate Scheme for the Cahn–Hilliard Ohta–Kawaski Phase-Field Model of Diblock Copolymer Melt Confined in Hele–Shaw Cell

Abstract

In this work, we consider numerical approximations of the phase-field model of diblock copolymer melt confined in Hele–Shaw cell, which is a very complicated coupled nonlinear system consisting of the Darcy equations and the Cahn–Hilliard type equations with the Ohta–Kawaski potential. Through the combination of a novel explicit-Invariant Energy Quadratization approach and the projection method, we develop the first full decoupling, energy stable, and second-order time-accurate numerical scheme. The introduction of two auxiliary variables and the design of two auxiliary ODEs play a vital role in obtaining the full decoupling structure while maintaining energy stability. The scheme is also linear and unconditional energy stable, and the practical implementation efficiency is also very high because it only needs to solve a few elliptic equations with constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Numerical experiments further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

Keywords

Phase-field / Explicit-IEQ / Darcy / Decoupled / Energy Stability / Diblock copolymer melt

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Funding

National Science Foundation(2012490)