Energy Stable Semi-implicit Schemes for Allen–Cahn–Ohta–Kawasaki Model in Binary System

Abstract

In this paper, we propose a first order energy stable linear semi-implicit method for solving the Allen-Cahn-Ohta-Kawasaki equation. By introducing a new nonlinear term in the Ohta-Kawasaki free energy functional, all the system forces in the dynamics are localized near the interfaces which results in the desire hyperbolic tangent profile. In our numerical method, the time discretization is done by some stabilization technique in which some extra nonlocal but linear term is introduced and treated explicitly together with other linear terms, while other nonlinear and nonlocal terms are treated implicitly. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are proved for this method in both semi-discretization and full discretization levels. Numerical experiments indicate the force localization and desire hyperbolic tangent profile due to the new nonlinear term. We test the first order temporal convergence rate of the proposed scheme. We also present hexagonal bubble assembly as one type of equilibrium for the Ohta-Kawasaki model. Additionally, the two-third law between the number of bubbles and the strength of long-range interaction is verified which agrees with the theoretical studies.

Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

“…We discretize the spatial operators using the spectral collocation approximation. To this end, we adopt some notations for the spectral approximation as in [20,10,37].…”

Two efficient block preconditioners for the mass-conserved Ohta-Kawasaki equation

“…One is the implicitly nonlinear term that will take the most time-consuming part in each time update. The common approaches to updating the nonlinear term include the Picard method [13,14], the Newton method [15,16], the nonlinear multigrid method [17,18], the preconditioned steepest descent algorithm [19,20], etc. The Picard method and the Newton method have been applied to the Ohta-Kawasaki equation [15,21].…”

Analysis for Allen-Cahn-Ohta-Nakazawa Model in a Ternary System

“…2) is adopted to mimic φ i , as the indicator for the A and B species, respectively. In our earlier work [3,4], a similar term has been introduced to some binary system with long-range interaction in order to study the associated L 2 gradient flow dynamics and maintain a better hyperbolic tangent profile for the solution and preserve its maximum principle at both continuous and discrete level. Meanwhile, we impose as usual fixed volume constraints…”