Energy stability of the temporal nonuniform fractional BDF2 scheme for the time fractional swift-Hohenberg equation

This paper introduces a high-order nonuniform finite difference scheme for the time-fractional Swift-Hohenberg equation, leveraging a nonuniform fractional BDF2 formula. The scheme’s unique solvability is established via the Brouwer theorem. By exploiting the discrete gradient structure of the fractional BDF2 formula, the energy stability of the difference scheme is theoretically derived. Numerical experiments validate the scheme’s accuracy and confirm the consistency between the discrete energy dissipation behavior and theoretical predictions.