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.
Keywords: Time-fractional Allen–Cahn equation; energy dissipation; L1 approximation; L2 approximation.