A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations (Communications in Computational Physics, 2023)
In this article, we study the energy dissipation property of time-fractionalAllen–Cahn equation. On the continuous level, we propose an upper bound of energythat decreases with respect to time and coincides with the original energy at $t=0$ andas $t$ tends to $ ∞.$ This upper bound can also be viewed as a nonlocal-in-time modifiedenergy which is the summation of the original energy and an accumulation term dueto the memory effect of time-fractional derivative. In particular, the decrease of themodified energy indicates that the original energy indeed decays w.r.t. time in a smallneighborhood at $t=0.$ We illustrate the theory mainly with the time-fractional Allen–Cahn equation but it could also be applied to other time-fractional phase-field modelssuch as the Cahn–Hilliard equation. On the discrete level, the decreasing upper boundof energy is useful for proving energy dissipation of numerical schemes. First-order L1and second-order L2 schemes for the time-fractional Allen–Cahn equation have similardecreasing modified energies, so that stability can be established. Some numericalresults are provided to illustrate the behavior of this modified energy and to verify ourtheoretical results.