Efficient numerical scheme for a penalized Allen–Cahn type Ohta–Kawasaki phase-field model for diblock copolymers

Abstract

In this paper, we develop an unconditionally energy stable, second-order accurate time marching scheme for solving a penalized Allen–Cahn type Ohta–Kawasaki phase-field model for diblock copolymers, where the total free energy of the system consists of the double-well potential, the nonlocal Ohta–Kawasaki free energy functional, and a penalization potential to enforce the conservation of the modified volume approximately. The developed scheme combines the SAV (scalar auxiliary variable) approach with the stabilization technique, where a crucial linear stabilization term is added to enhance the stability while using the large time steps. The scheme is very easy-to-implement and one only needs to solve two decoupled elliptic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously and demonstrate the stability and the accuracy of the developed scheme numerically through simulating numerous numerical examples in 2D and 3D.