Year: 2022
Author: Kelong Cheng, Cheng Wang, Steven M. Wise, Yanmei Wu
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 2 : pp. 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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2021-0165
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 2 : pp. 279–303
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
Keywords: Cahn-Hilliard equation third order backward differentiation formula unique solvability energy stability discrete $L^6_N$ estimate optimal rate convergence analysis.
Author Details
-
A Novel Energy-Optimized Technique of SAV-Based (EOP-SAV) Approaches for Dissipative Systems
Liu, Zhengguang | Zhang, Yanrong | Li, XiaoliJournal of Scientific Computing, Vol. 101 (2024), Iss. 2
https://doi.org/10.1007/s10915-024-02677-0 [Citations: 2] -
A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility
Hou, Dianming | Ju, Lili | Qiao, ZhonghuaMathematics of Computation, Vol. 92 (2023), Iss. 344 P.2515
https://doi.org/10.1090/mcom/3843 [Citations: 8] -
The high-order exponential semi-implicit scalar auxiliary variable approach for the general nonlocal Cahn-Hilliard equation
Meng, Xiaoqing | Cheng, Aijie | Liu, ZhengguangCommunications in Nonlinear Science and Numerical Simulation, Vol. 137 (2024), Iss. P.108169
https://doi.org/10.1016/j.cnsns.2024.108169 [Citations: 1] -
A third-order positivity-preserving and energy stable numerical scheme for the Cahn-Hilliard equation with logarithmic potential
Yuhuan, Li | Jianyu, Jing | Qianqian, Liu | Cheng, Wang | Wenbin, ChenSCIENTIA SINICA Mathematica, Vol. (2024), Iss.
https://doi.org/10.1360/SSM-20223-0014 [Citations: 0] -
Efficient and energy stable numerical schemes for the two-mode phase field crystal equation
Zhang, Fan | Li, Dongfang | Sun, Hai-WeiJournal of Computational and Applied Mathematics, Vol. 427 (2023), Iss. P.115148
https://doi.org/10.1016/j.cam.2023.115148 [Citations: 5] -
Stability and Error Estimates of High Order BDF-LDG Discretizations for the Allen–Cahn Equation
Yan, Fengna | Cheng, ZiqiangComputational Mathematics and Mathematical Physics, Vol. 63 (2023), Iss. 12 P.2551
https://doi.org/10.1134/S0965542523120229 [Citations: 0] -
A-Stable High-Order Block Implicit Methods for Parabolic Equations
Li, Shishun | Wang, Jing-Yuan | Cai, Xiao-ChuanSIAM Journal on Numerical Analysis, Vol. 61 (2023), Iss. 4 P.1858
https://doi.org/10.1137/22M152880X [Citations: 0] -
Phase-field modeling and linearly energy-stable Runge–Kutta algorithm of colloidal crystals on curved surfaces
Yang, Junxiang | Li, Yibao | Kim, JunseokJournal of Computational and Applied Mathematics, Vol. 443 (2024), Iss. P.115750
https://doi.org/10.1016/j.cam.2023.115750 [Citations: 1] -
A variational Crank–Nicolson ensemble Monte Carlo algorithm for a heat equation under uncertainty
Ye, Changlun | Yao, Tingfu | Bi, Hai | Luo, XianbingJournal of Computational and Applied Mathematics, Vol. 451 (2024), Iss. P.116068
https://doi.org/10.1016/j.cam.2024.116068 [Citations: 0] -
New third-order convex splitting methods and analysis for the phase field crystal equation
Ye, Zhijian | Zheng, Zhoushun | Li, ZhilinNumerical Algorithms, Vol. (2024), Iss.
https://doi.org/10.1007/s11075-024-01782-3 [Citations: 0] -
Consistently and unconditionally energy-stable linear method for the diffuse-interface model of narrow volume reconstruction
Yang, Junxiang | Kim, JunseokEngineering with Computers, Vol. 40 (2024), Iss. 4 P.2617
https://doi.org/10.1007/s00366-023-01935-3 [Citations: 1] -
Second-order energy-stable scheme and superconvergence for the finite difference method on non-uniform grids for the viscous Cahn–Hilliard equation
Chen, Yanping | Yan, Yujing | Li, Xiaoli | Zhao, XuanCalcolo, Vol. 61 (2024), Iss. 2
https://doi.org/10.1007/s10092-024-00579-z [Citations: 0] -
A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear Term
Cui, Ming | Niu, Yiyi | Xu, ZhenJournal of Scientific Computing, Vol. 99 (2024), Iss. 1
https://doi.org/10.1007/s10915-024-02490-9 [Citations: 1] -
Efficient and unconditionally energy stable exponential-SAV schemes for the phase field crystal equation
Zhang, Fan | Sun, Hai-Wei | Sun, TaoApplied Mathematics and Computation, Vol. 470 (2024), Iss. P.128592
https://doi.org/10.1016/j.amc.2024.128592 [Citations: 1] -
A family of structure-preserving exponential time differencing Runge–Kutta schemes for the viscous Cahn–Hilliard equation
Sun, Jingwei | Zhang, Hong | Qian, Xu | Song, SongheJournal of Computational Physics, Vol. 492 (2023), Iss. P.112414
https://doi.org/10.1016/j.jcp.2023.112414 [Citations: 5]