TY - JOUR
T1 - A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation
AU - Yue Yan, Wenbin Chen, Cheng Wang & Steven M. Wise
JO - Communications in Computational Physics
VL - 2
SP - 572
EP - 602
PY - 2018
DA - 2018/02
SN - 23
DO - http://doi.org/10.4208/cicp.OA-2016-0197
UR - https://global-sci.org/intro/article_detail/cicp/10539.html
KW - Cahn-Hilliard equation, energy stable BDF, Douglas-Dupont regularization, mixed finite element, energy stability.
AB - <p style="text-align: justify;">In this paper we present a second order accurate (in time) energy stable numerical
scheme for the Cahn-Hilliard (CH) equation, with a mixed finite element approximation
in space. Instead of the standard second order Crank-Nicolson methodology,
we apply the implicit backward differentiation formula (BDF) concept to derive
second order temporal accuracy, but modified so that the concave diffusion term is
treated explicitly. This explicit treatment for the concave part of the chemical potential
ensures the unique solvability of the scheme without sacrificing energy stability. An
additional term $A$τ∆($u^{k+1}$−$u^k$) is added, which represents a second order Douglas-Dupont-type
regularization, and a careful calculation shows that energy stability is
guaranteed, provided the mild condition $A$≥$\frac{1}{16}$<span style="font-family: 微软雅黑; color: #333333; font-size: 14px; background: #FFFFFF;">&nbsp;</span>is enforced. In turn, a uniform in time $H^1$ bound of the numerical solution becomes available. As a result, we are able to
establish an $ℓ^∞$(0,$T$;$L^2$) convergence analysis for the proposed fully discrete scheme,
with full&nbsp;$\mathcal{O}$<span style="font-family: Arial; font-size: 14px;">&nbsp;</span>(τ<sup>2</sup>+$h^2$) accuracy. This convergence turns out to be unconditional; no scaling
law is needed between the time step size $τ$ and the spatial grid size $h$. A few
numerical experiments are presented to conclude the article.</p>