Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model

Abstract

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 $H^{-\alpha }$, $\alpha \in (0,1)$. 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 scale evolutions of the solution in simulations.

Funding

National Natural Science Foundation of China(11861060); State Key Program of National Natural Science Foundation of China(61833005); ZhiShan Youth Scholar Program of SEU