A fully implicit numerical scheme is established for solving the time fractional Swift-Hohenberg (TFSH) equation with a Caputo time derivative of order $\alpha\in(0,1)$. The variable-step L1 formula and the finite difference method are employed for the time and the space discretizations, respectively. The unique solvability of the numerical scheme is proved by the Brouwer fixed-point theorem. With the help of the discrete convolution form of L1 formula, the time-stepping scheme is shown to preserve a discrete energy dissipation law which is asymptotically compatible with the classic energy law as $\alpha\to1^-$. Furthermore, the $L^\infty$ norm boundedness of the discrete solution is obtained. Combining with the global consistency error analysis framework, the $L^2$ norm convergence order is shown rigorously. Several numerical examples are provided to illustrate the accuracy and the energy dissipation law of the proposed method. In particular, the adaptive time-stepping strategy is utilized to capture the multi-scale time behavior of the TFSH model efficiently.

翻译：针对Caputo时间导数为 $\alpha\in(0,1)$ 阶的时间分数阶Swift-Hohenberg (TFSH) 方程，本文建立了一种全隐式数值格式。时间离散采用变步长L1公式，空间离散采用有限差分方法。通过Brouwer不动点定理证明了数值格式的惟一可解性。借助L1公式的离散卷积形式，证明该时间步进格式保持了离散能量耗散律，且当 $\alpha\to1^-$ 时，该耗散律与经典能量律渐近兼容。此外，得到了离散解的 $L^\infty$ 范数有界性。结合全局一致性误差分析框架，严格证明了 $L^2$ 范数收敛阶。最后通过多个数值算例展示了所提方法的精度与能量耗散律。特别地，采用自适应时间步进策略有效捕捉了TFSH模型的多尺度时间行为。