@Article{NMTMA-15-279,
author = {Cheng , KelongWang , ChengWise , Steven M. and Wu , Yanmei},
title = {A Third Order Accurate in Time, BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2022},
volume = {15},
number = {2},
pages = {279--303},
abstract = {<p style="text-align: justify;">In this paper we propose and analyze a backward differentiation formula
(BDF) type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy. The Fourier pseudo-spectral method is used to discretize space. The
surface diffusion and the nonlinear chemical potential terms are treated implicitly,
while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont
regularization term, in the form of $−A_0\Delta t^2\Delta_N (\phi^{n+1}−\phi^n),$ is added in the numerical
scheme. In particular, the energy stability is carefully derived in a modified version,
so that a uniform bound for the original energy functional is available, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy
stability analysis, a uniform-in-time $L^6_N$ bound of the numerical solution is obtained.
And also, the optimal rate convergence analysis and error estimate are provided, in
the $L^∞_{∆t} (0, T ;L^2 _N) ∩ L^2_{∆t} (0,T; H^2_h)$ norm, with the help of the $L^6_N$ bound for the numerical solution. A few numerical simulation results are presented to demonstrate
the efficiency of the numerical scheme and the third order convergence.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2021-0165},
url = {http://global-sci.org/intro/article_detail/nmtma/20353.html}
}