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

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+1ϕn), 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 Lt(0,T;L2N)L2t(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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