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

Abstract: 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.