In this paper, we propose and analyze a linear, structure-preserving method for solving the Allen–Cahn equation based on the second-order backward differentiation formula (BDF2) with variable time steps and the scalar auxiliary variable (SAV) approach. 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) enabling the development of effective stabilization terms that facilitate the construction of maximum bound principle (MBP)-preserving linear methods. Combining this novel functional with a standard central difference stencil, we then propose a linear, second-order variable-step BDF2 type stabilized exponential SAV scheme, referred to as BDF2-sESAV-I. The scheme is proven 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, the approximation of the original energy by the modified energy is analyzed. By leveraging the kernel recombination technique, optimal \( H^{1}\) - and \( L^{\infty }\) -norm error estimates for 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.
