Energy stable L2 schemes for time-fractional phase-field equations

Abstract

In this work, we establish the energy stability of high-order L2-type schemes for time-fractional phase-field equations. We propose a reformulation of the discrete L2 operator, show the monotonicity of some associated coefficients, and then obtain two positive definiteness properties for the L2 operator. Based on these two properties, we show that the energy is bounded by the initial energy for the L2 scalar auxiliary variable schemes of time-fractional gradient flows. Furthermore, a fractional energy law can be established for the L2 implicit-explicit scheme of the time-fractional Allen–Cahn equation. Several numerical experiments are provided to verify the stability as well as the convergence.