TY - JOUR
T1 - The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation
AU - Liao , Hong-Lin
AU - Zhu , Xiaohan
AU - Wang , Jindi
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 1128
EP - 1146
PY - 2022
DA - 2022/10
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2022-0011s
UR - https://global-sci.org/intro/article_detail/nmtma/21096.html
KW - Time-fractional Allen-Cahn equation, adaptive L1 scheme, variational energy dissipation law, orthogonal convolution kernels, complementary convolution kernels.
AB - <p style="text-align: justify;">In this work, we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form. The L1 implicit
scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools, i.e., the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then the
discrete embedding techniques and the fractional Grönwall inequality are applied
to establish an $L^2$ norm error estimate on nonuniform time meshes. An adaptive
time-stepping strategy according to the dynamical feature of the system is presented
to capture the multi-scale behaviors and to improve the computational performance.</p>