Maximum bound principle preserving exponential time differencing schemes for the penalized Allen-Cahn-Ohta-Kawasaki model

The nonlocal penalized Allen-Cahn-Ohta-Kawasaki (pACOK) equation,equipped with a specific nonlinear function in the Ohta-Kawasaki free energy functional, satisfies the maximum bound principle (MBP).In this paper, we develop and analyze the first- and second-order exponential time differencing (ETD) schemes for solving the pACOK equation.The fully discrete numerical schemes are obtained by adopting the second-order central difference approximation in space.This is an initial exploration of the application of the ETD schemes on the pACOK equation and, to the best of our knowledge,also the first work on the second-order linear scheme for this model with the complete theoretical proofs of the MBP preservation and unconditional energy stability.In addition, numerical experiments convincingly demonstrate superior accuracy and faster convergence rates of the proposed ETD schemescompared with the existing stabilized linear schemes. Furthermore, we simulate various microstructures of diblock copolymers.