In this paper, we present an efficient and energy-stable scheme based on the Crank–Nicolson formula for the modified phase-field crystal equation with a strong nonlinear vacancy potential. In the scheme, the nonlinear terms (the first derivatives of the double-well and vacancy potentials) are treated explicitly, which makes the scheme efficient, and the energy stability is guaranteed by assuming that the second derivatives of the double-well and vacancy potentials are each bounded and by adding two second-order stabilization terms. In particular, by bounding the second derivatives of the double-well and vacancy potentials, respectively, we can choose the stabilization parameters independently of the vacancy parameter. As a result, the convergence constant and energy decay trend are not affected by the vacancy parameter.

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.

The slope-selection epitaxial thin-film growth model was formulated as the $L^2$-gradient flow of an energy functional with a nonlinear potential of the surface slope. We developed a second-order, linear, and energy-stable scheme for this model based on a linear convex splitting in which the nonlinear potential term was treated explicitly together with an auxiliary term that ensured the convexity of the explicit part. The scheme was constructed using a second-order strong-stability-preserving implicit–explicit Runge–Kutta method. We explicitly identified the range of Runge–Kutta coefficients for which the original discrete energy decay property held, and proved that the scheme was unconditionally energy-stable with respect to the original discrete energy functional. Numerical results were presented to verify the accuracy, computational efficiency, and long-time energy stability of the scheme.