A Third Order Accurate in Time, BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation
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 −A0Δt2ΔN(ϕn+1−ϕn), 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 L6N 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;L2N)∩L2∆t(0,T;H2h) norm, with the help of the L6N 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.
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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 L6N estimate optimal rate convergence analysis.
Author Details
Kelong Cheng Email
Cheng Wang Email
Steven M. Wise Email
Yanmei Wu Email
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