TY - JOUR
T1 - Numerical Energy Dissipation for Time-Fractional Phase-Field Equations
AU - Quan , Chaoyu
AU - Tang , Tao
AU - Yang , Jiang
JO - Journal of Computational Mathematics
VL - 3
SP - 515
EP - 539
PY - 2024
DA - 2024/11
SN - 43
DO - http://doi.org/10.4208/jcm.2311-m2021-0199
UR - https://global-sci.org/intro/article_detail/jcm/23548.html
KW - Time-fractional phased-field equation, Allen-Cahn equations, Cahn-Hilliard
equations, Caputo fractional derivative, Energy dissipation.
AB - <p style="text-align: justify;">The numerical integration of phase-field equations is a delicate task which needs to
recover at the discrete level intrinsic properties of the solution such as energy dissipation
and maximum principle. Although the theory of energy dissipation for classical phase field
models is well established, the corresponding theory for time-fractional phase-field models
is still incomplete. In this article, we study certain nonlocal-in-time energies using the
first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete
fractional energy law and a discrete weighted energy law. The extension for a $(2−α)$-order
L1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate that
the energy bound is preserved for the L1 schemes with nonuniform time steps. Several
numerical experiments are carried to verify our theoretical analysis.</p>