Stability and convergence analysis of adaptive BDF2 scheme for the Swift-Hohenberg equation

The design, analysis and numerical simulations of a stabilized variable time-stepping difference scheme, for the Swift–Hohenberg equation, are considered in this paper. The proposed time-stepping scheme is proved to preserve a discrete energy dissipation law. With the help of the new discrete orthogonal convolution kernels, the unique solvability and the unconditional energy stability of the numerical scheme are rigorously proved. In addition, the second-order L 2 norm convergence both in time and in space, of the proposed scheme, is shown under the almost independent of the time-step ratios. To the best of our knowledge, this is the first time that the L 2 norm convergence of the adaptive BDF2 method is achieved for the Swift–Hohenberg equation. Numerical examples are provided to illustrate our theoretical results and to show the computational efficiency of the numerical scheme. • A stabilized variable time-stepping scheme for the Swift-Hohenberg equation is developed. • The theoretical results of the time-stepping scheme are rigorously proved. • The proposed adaptive time-stepping algorithms are suitable for the long time simulation.