A novel adaptive Crank–Nicolson-type scheme for the time fractional Allen–Cahn model

Abstract

An energy stable Crank–Nicolson-type scheme with unequal time-steps is proposed for time fractional Allen–Cahn model. The new scheme is build upon a reformulated problem associated with the Riemann–Liouville fractional derivative and the so-called L1R formula from Tang et al. (2021). With the standard trapezoidal approximation of double-well potential, it is shown to be stable in a variational energy sense by using a novel discrete convolution inequality (discrete gradient structure) of the L1R formula. The resulting discrete energy dissipation law asymptotically also preserves the classical energy dissipation law when the fractional order α→1. Numerical examples with an adaptive time-stepping strategy are included to show the effectiveness of the proposed scheme.