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Volume 15, Issue 2
A Third Order Accurate in Time, BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation

Kelong Cheng, Cheng Wang, Steven M. Wise & Yanmei Wu

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 279-303.

Published online: 2022-03

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  • Abstract

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.

  • AMS Subject Headings

35K30, 35K55, 65L06, 65M12, 65M70, 65T40

  • Copyright

COPYRIGHT: © Global Science Press

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@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 = {

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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0165}, url = {http://global-sci.org/intro/article_detail/nmtma/20353.html} }
TY - JOUR T1 - A Third Order Accurate in Time, BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation AU - Cheng , Kelong AU - Wang , Cheng AU - Wise , Steven M. AU - Wu , Yanmei JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 279 EP - 303 PY - 2022 DA - 2022/03 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2021-0165 UR - https://global-sci.org/intro/article_detail/nmtma/20353.html KW - Cahn-Hilliard equation, third order backward differentiation formula, unique solvability, energy stability, discrete $L^6_N$ estimate, optimal rate convergence analysis. AB -

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.

Cheng , KelongWang , ChengWise , Steven M. and Wu , Yanmei. (2022). A Third Order Accurate in Time, BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation. Numerical Mathematics: Theory, Methods and Applications. 15 (2). 279-303. doi:10.4208/nmtma.OA-2021-0165
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