TY - JOUR
T1 - A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations
AU - Quan , Chaoyu
AU - Tang , Tao
AU - Wang , Boyi
AU - Yang , Jiang
JO - Communications in Computational Physics
VL - 4
SP - 962
EP - 991
PY - 2023
DA - 2023/05
SN - 33
DO - http://doi.org/10.4208/cicp.OA-2022-0148
UR - https://global-sci.org/intro/article_detail/cicp/21666.html
KW - Time-fractional Allen–Cahn equation, energy dissipation, L1 approximation, L2 approximation.
AB - <p style="text-align: justify;">In this article, we study the energy dissipation property of time-fractional
Allen–Cahn equation. On the continuous level, we propose an upper bound of energy
that decreases with respect to time and coincides with the original energy at $t=0$ and
as $t$ tends to $ ∞.$ This upper bound can also be viewed as a nonlocal-in-time modified
energy which is the summation of the original energy and an accumulation term due
to the memory effect of time-fractional derivative. In particular, the decrease of the
modified energy indicates that the original energy indeed decays w.r.t. time in a small
neighborhood 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 models
such as the Cahn–Hilliard equation. On the discrete level, the decreasing upper bound
of energy is useful for proving energy dissipation of numerical schemes. First-order L1
and second-order L2 schemes for the time-fractional Allen–Cahn equation have similar
decreasing modified energies, so that stability can be established. Some numerical
results are provided to illustrate the behavior of this modified energy and to verify our
theoretical results.</p>