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A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation

Author: Zhi Zhong Sun
Journal: Math. Comp. 64 (1995), 1463-1471
MSC: Primary 65M06; Secondary 65M12
MathSciNet review: 1308465
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Abstract: The Cahn-Hilliard equation is a nonlinear evolutionary equation that is of fourth order in space. In this paper a linearized finite difference scheme is derived by the method of reduction of order. It is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete $ {L_2}$-norm. The coefficient matrix of the difference system is symmetric and positive definite, so many well-known iterative methods (e.g. Gauss-Seidel, SOR) can be used to solve the system.

References [Enhancements On Off] (What's this?)

  • [1] A. Novick-Cohen and L.A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Phys.D 10 (1984), 277-298. MR 763473 (85k:35120)
  • [2] C.M. Elliott and D. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math. 38 (1987), 97-128. MR 983721 (90f:80004)
  • [3] -, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal. 26 (1989), 884-903. MR 1005515 (90k:65163)
  • [4] C.M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp. 58 (1992), 603-630. MR 1122067 (92f:65116)
  • [5] Sun Zhi-zhong, A new class of difference schemes for linear parabolic differential equations, Math. Numer. Sinica 16 (1994), 115-130 (in Chinese); Chinese J. Numer. Math. Appl. 16 (1994), No. 3, 1-20.
  • [6] -, The method of the reduction of order for the numerical solution to elliptic differential equations, J. Southeast Univ. 23 (1993), No. 6, 8-16. (in Chinese) MR 1286873 (95e:65093)
  • [7] -, A class of second-order accurate difference schemes for quasi-linear parabolic differential equations, Math. Numer. Sinica 16 (1994), 347-361. (in Chinese) MR 1393862 (97b:65102)

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Keywords: Cahn-Hilliard equation, nonlinear evolution equation, finite difference convergence, solvability
Article copyright: © Copyright 1995 American Mathematical Society

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