A fully discrete implicit scheme is proposed for the Swift-Hohenberg model, combining the third-order backward differentiation formula (BDF3) for the time discretization and the second-order finite difference scheme for the space discretization. Applying the Brouwer fixed-point theorem and the positive definiteness of the convolution coefficients of BDF3, the presented numerical algorithm is proved to be uniquely solvable and unconditionally energy stable, further, the numerical solution is shown to be bounded in the maximum norm. The proposed scheme is rigorously proved to be convergent in L 2 norm by the discrete orthogonal convolution (DOC) kernel, which transfer the four-level-solution form into the three-levelgradient form for the approximation of the temporal derivative. Consequently, the error estimate for the numerical solution is established by utilization of the discrete Gronwall inequality. Numerical examples in 2D and 3D cases are provided to support the theoretical results.

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An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H − α , α∈(0 ,1). The Fourier pseudo-spectral method is applied for the spatial approximation. The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model, primarily due to the introduction of the fractional Laplacian. This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian. Subsequently, the unique solvability and the modified energy dissipation law are theoretically guaranteed. We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term. Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian. In particular, we demonstrate that the adaptive time-stepping strategy, compared with the uniform time steps, captures the multiple time scales evolutions of the solution in simulations.