A high-efficiency second-order numerical scheme for time-fractional phase field models by using extended SAV method

Abstract

In this paper, a second-order numerical scheme for the time-fractional phase field models is proposed. In this scheme, the fractional backward difference formula is used to approximate the time-fractional derivative and the extended scalar auxiliary variable method is used to deal with the nonlinear terms. The energy dissipation property for the numerical scheme is proved. Our discussion includes the time-fractional Allen–Cahn equation, the time-fractional Cahn–Hilliard equation, and the time-fractional molecular beam epitaxy model. In the numerical implementation, a fast method based on a globally uniform approximation of the trapezoidal rule for the integral on the real line is adopted to decrease the memory requirement and computational cost. Finally, some numerical examples are given to confirm the effectiveness of the proposed methods.