High-order energy stable algorithm for time-fractional Swift-Hohenberg model on graded meshes

The time-fractional Swift-Hohenberg (TFSH) model, which is widely applied for describing various pattern formations, is considered in this work. We propose a fast high-order numerical algorithm for this model. To the best of our knowledge, this is the first (3-α)-order (α∈(0,1)) numerical method for solving the TFSH model with better temporal accuracy than existing numerical schemes. As solutions to time-fractional problems often display weak singularity near t=0, the graded mesh method is employed to construct numerical schemes with optimal convergence rates. Despite the theoretical challenges introduced by the nonlinearity and graded meshes, we prove that the numerical schemes are uniquely solvable. With the help of the discrete gradient structure (DGS), the energy stability analysis of the proposed scheme is also established. Since the nonlocal feature of the Caputo fractional derivative usually results in huge computational and storage costs, we apply the sum-of-exponentials (SOE) method to construct fast schemes, thereby improving computational efficiency. And the Fourier spectral method is employed for the spatial approximation. Finally, several numerical experiments are conducted to demonstrate the efficiency of the proposed algorithm in 2D and 3D space.