Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen–Cahn model

The main purpose of this paper is to solve the fractional non-local Allen–Cahn model via a super fast explicit operator splitting spectral (FEOSS) method, which is based on the Strang splitting scheme for time discretization and the Fourier spectral method for space discretization. For a problem of size N, this method reduces the amount of storage from O(N2) to O(N) and cuts down the computational cost from O(N3) to O(Nlog2N). Moreover, significant computational gains can be obtained by applying a new adaptive time-stepping algorithm based on the local error method. This allows accessing time-scales that vary by several orders of magnitude. Numerical results demonstrate that the FEOSS method with adaptive time-stepping algorithm achieves a remarkable resolution and accuracy in a very efficient manner.