TY - JOUR
T1 - Efficient High-Order Backward Difference Formulae for Cahn-Hilliard Equation with the Gradient Flow in $H^{−α}$
AU - Xue , Zhongqin
AU - Wen , Guanghui
AU - Zhang , Zhimin
AU - Zhao , Xuan
JO - Communications in Computational Physics
VL - 5
SP - 1263
EP - 1286
PY - 2024
DA - 2024/06
SN - 35
DO - http://doi.org/10.4208/cicp.OA-2023-0315
UR - https://global-sci.org/intro/article_detail/cicp/23191.html
KW - Cahn-Hilliard equation, high-order BDF methods, discrete gradient structure, energy stability, convergence analysis.
AB - <p style="text-align: justify;">High-order schemes enable simulating multi-time-scale problems with relatively large time step sizes for fairly accurate solutions. An unified high-order implicit BDF-$q$ $(q = 3,4,5)$ scheme is developed for the Cahn-Hilliard equation with the
gradient flow in $H^{−α},α∈(0,1],$ including the classical case and its fractional variants.
Introducing the discrete gradient structure, the resulting implicit BDF-$q$ scheme is presented to satisfy the discrete energy dissipation law, which is actually compatible with
the one for the classical case as the order of the fractional Laplacian $α$ approaches 1.
The $L^2$ norm error estimate for the BDF-$q$ scheme is rigorously proved by developing
a discrete Young-type convolution inequality to deal with the nonlinear term along
with the fractional Laplacian. Further, the high-order BDF-$q$ scheme is shown to be
less time-consuming compared to the variable-step BDF-2 scheme, while the BDF-5
scheme reduces the CPU time in long-time simulation of coarsening dynamics by almost 80%. Numerical examples also demonstrate that high-order schemes are deemed
appealing for long-time slow evolution, while variable-step scheme exhibits more flexibility during phase separation at initial state. In light of this, the variable-step BDF-$q$ scheme utilizing the adaptive time-stepping strategy is implemented to capture both
the rapid and slow evolutions of the solutions efficiently and accurately even in high
dimensions.</p>