Communications in Mathematical Sciences

Volume 11 (2013)

Number 2

On the unconditionally gradient stable scheme for the Cahn-Hilliard equation and its implementation with Fourier method

Pages: 345 – 360

DOI: https://dx.doi.org/10.4310/CMS.2013.v11.n2.a1

Authors

Andrew Christlieb (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Keith Promislow (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Zhengfu Xu (Department of Mathematical Science, Michigan Technological University, Houghton, Mich., U.S.A.)

Abstract

We implement a nonlinear unconditionally gradient stable scheme by Eyre, within the Fourier method framework for the long-time numerical integration of the Allen-Cahn and Cahn-Hilliard equations, which are gradient flows of the Allen-Cahn energy. We propose a new iterative procedure to solve the nonlinear scheme. When the iterative scheme is applied to the Allen-Cahn equation, we show that the nonlinear iteration is a contractive mapping in the $L^2$ norm for large time steps. For the Cahn-Hilliard equation, we establish that the proposed iterative scheme converges with a time step constraint. Further, we numerically demonstrate that the iterative scheme converges for large time steps. The scheme allows for spectral accuracy in space and fast simulation of the dynamics in high dimensions while preserving the discrete form of the energy law. For the general potential well, we present the gradient stable scheme without introducing extra stabilizing terms. Therefore, the numerical error introduced by the operator splitting is reduced to its minimum.

Keywords

Cahn-Hilliard equation, phase-field model, unconditionally gradient stable scheme, Fourier method, iterative method

2010 Mathematics Subject Classification

65T50, 82C26

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Published 7 December 2012