Second-order linear adaptive time-stepping schemes for the fractional Allen–Cahn equation.

In this paper, two second-order stable schemes based on the scalar auxiliary variable (SAV) approach are constructed for the fractional-in-space Allen–Cahn equation. We use the Crank–Nicolson and BDF2 schemes for time discretization, and prove the unique solvability and energy stability. The fully discrete schemes are carried out with the Fourier spectral approximation in space. Finally, we provide numerical examples to verify the accuracy of the proposed methods. In particular, we develop an adaptive time-stepping approach related to the accuracy requirement to speed up the calculation. [ABSTRACT FROM AUTHOR]