A mass conservative and energy stable scheme for the conservative Allen–Cahn type Ohta–Kawasaki model for diblock copolymers

The conservative Allen–Cahn type Ohta–Kawasaki model was introduced to reformulate the Cahn–Hilliard type Ohta–Kawasaki model. A difficulty in numerically solving the conservative Allen–Cahn type Ohta–Kawasaki model is how to discretize the nonlinear and nonlocal terms in time to preserve the mass conservation and energy decay properties without losing the efficiency and accuracy. To settle this problem, we present a linear, second-order, mass conservative, and energy stable scheme based on the Crank–Nicolson formula. In the scheme, the nonlinear and nonlocal terms are explicitly treated, which make the scheme linear, and the energy stability is guaranteed by adopting a truncated double-well potential and by adding two second-order stabilization terms. We analytically and numerically show that the scheme is mass conservative and energy stable. Additionally, the scheme can be easily implemented within a few lines of MATLAB code.