A unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn-Hilliard equation on generalized SAV approach

An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn-Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., \Ass{1}: $0<r_k:=\tau_k/\tau_{k-1}< r_{\max}\approx 4.8645$, we establish a rigorous error estimate in $H^1$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^1$-norm. The $H^1$ bound of the numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.