  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.
