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A fully implicit numerical scheme is established for solving the timefractional Swift-Hohenberg (TFSH) equation with a Caputo time derivative oforder $\alpha\in(0,1)$. The variable-step L1 formula and the finite differencemethod are employed for the time and the space discretizations, respectively.The unique solvability of the numerical scheme is proved by the Brouwerfixed-point theorem. With the help of the discrete convolution form of L1formula, the time-stepping scheme is shown to preserve a discrete energydissipation law which is asymptotically compatible with the classic energy lawas $\alpha\to1^-$. Furthermore, the $L^\infty$ norm boundedness of the discretesolution is obtained. Combining with the global consistency error analysisframework, the $L^2$ norm convergence order is shown rigorously. Severalnumerical examples are provided to illustrate the accuracy and the energydissipation law of the proposed method. In particular, the adaptivetime-stepping strategy is utilized to capture the multi-scale time behavior ofthe TFSH model efficiently.

Energy stability and convergence of variable-step L1 scheme for the time fractional Swift-Hohenberg model