Energy dissipation law and maximum bound principle-preserving linear BDF2 schemes with variable steps for the Allen-Cahn equation

In this paper, we propose and analyze a linear, structure-preserving scalar auxiliary variable (SAV) method for solving the Allen--Cahn equation based on the second-order backward differentiation formula (BDF2) with variable time steps. To this end, we first design a novel and essential auxiliary functional that serves twofold functions: (i) ensuring that a first-order approximation to the auxiliary variable, which is essentially important for deriving the unconditional energy dissipation law, does not affect the second-order temporal accuracy of the phase function $\phi$; and (ii) allowing us to develop effective stabilization terms that are helpful to establish the MBP-preserving linear methods. Together with this novel functional and standard central difference stencil, we then propose a linear, second-order variable-step BDF2 type stabilized exponential SAV scheme, namely BDF2-sESAV-I, which is shown to preserve both the discrete modified energy dissipation law under the temporal stepsize ratio $0 < r_{k} := \tau_{k}/\tau_{k-1} < 4.864 - \delta$ with a positive constant $\delta$ and the MBP under $0 < r_{k} < 1 + \sqrt{2}$. Moreover, an analysis of the approximation to the original energy by the modified one is presented. With the help of the kernel recombination technique, optimal $H^{1}$- and $L^{\infty}$-norm error estimates of the variable-step BDF2-sESAV-I scheme are rigorously established. Numerical examples are carried out to verify the theoretical results and demonstrate the effectiveness and efficiency of the proposed scheme.