Volume 23, Issue 2
A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation

Commun. Comput. Phys., 23 (2018), pp. 572-602.

Published online: 2018-02

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• Abstract

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 $A$≥$\frac{1}{16}$ 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 $\mathcal{O}$ 2+$h^2$) 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.

• Keywords

Cahn-Hilliard equation, energy stable BDF, Douglas-Dupont regularization, mixed finite element, energy stability.

35K35, 35K55, 65M12, 65M60

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@Article{CiCP-23-572, author = {}, title = {A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation}, journal = {Communications in Computational Physics}, year = {2018}, volume = {23}, number = {2}, pages = {572--602}, abstract = {

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 $A$≥$\frac{1}{16}$ 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 $\mathcal{O}$ 2+$h^2$) 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0197}, url = {http://global-sci.org/intro/article_detail/cicp/10539.html} }
TY - JOUR T1 - A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation 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 -

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 $A$≥$\frac{1}{16}$ 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 $\mathcal{O}$ 2+$h^2$) 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.

Yue Yan, Wenbin Chen, Cheng Wang & Steven M. Wise. (2020). A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation. Communications in Computational Physics. 23 (2). 572-602. doi:10.4208/cicp.OA-2016-0197
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