Energy Dissipation Law of the Temporal Variable-Step Fractional BDF2 Scheme for Space–Time-Fractional Cahn–Hilliard Equation

Abstract

A high-order variable-step numerical scheme is formulated for the space–time-fractional Cahn–Hilliard equation, employing the variable-step fractional BDF2 formula. The unique solvability and mass conservation at the discretization setting are established. Subject to the constraint of time-step ratios, i.e., 0.4159≤rk≤4.660, a careful analysis based on the discrete gradient structure of the fractional BDF2 formula reveals that the proposed scheme adheres to the energy dissipation law. Remarkably, the modified energy exhibits asymptotic compatibility with that of the classical Cahn–Hilliard equation. Moreover, the modified energy dissipation law of the resulting scheme for the space–time-fractional Cahn–Hilliard equation aligns asymptotically with that of the variable-step BDF2 scheme for its classical counterpart. Finally, a few numerical experiments combined with the adaptive method are presented, which confirm the accuracy and efficiency of the proposed scheme.

Keywords

• space–time-fractional Cahn–Hilliard equation
• fractional BDF2 formula
• energy dissipation law
• asymptotically compatible