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.

翻译：本文针对Allen-Cahn方程，基于变步长二阶向后微分公式（BDF2），提出并分析了一种线性结构保持的标量辅助变量（SAV）方法。为此，我们首先设计了一个新颖且本质的辅助泛函，该泛函具有双重功能：（i）确保对辅助变量的一阶逼近（这对推导无条件能量耗散律至关重要）不会影响相函数$\phi$的二阶时间精度；（ii）使我们能够构造有效的稳定项，从而有助于建立保持最大有界原理（MBP）的线性方法。结合这一新颖泛函与标准中心差分格式，我们进而提出了一种线性、二阶变步长BDF2型稳定化指数SAV格式，记为BDF2-sESAV-I。该格式被证明在时间步长比$0 < r_{k} := \tau_{k}/\tau_{k-1} < 4.864 - \delta$（其中$\delta$为正常数）条件下保持离散修正能量耗散律，并在$0 < r_{k} < 1 + \sqrt{2}$条件下保持最大有界原理。此外，本文分析了修正能量对原始能量的逼近性质。借助核重组技术，我们严格建立了变步长BDF2-sESAV-I格式在$H^{1}$范数与$L^{\infty}$范数下的最优误差估计。数值算例验证了理论结果，并证明了所提格式的有效性与计算效率。