An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation

We present an unconditionally stable second-order hybrid numerical method for solving the Allen-Cahn equation representing a model for antiphase domain coarsening in a binary mixture. The proposed method is based on operator splitting techniques. The Allen-Cahn equation was divided into a linear and a nonlinear equation. First, the linear equation was discretized using a Crank-Nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method. The nonlinear equation was then solved analytically due to the availability of a closed-form solution. Various numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the scheme is unconditionally stable and second-order accurate in both time and space.

[1]  Andreas Prohl,et al.  Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows , 2003, Numerische Mathematik.

[2]  Jie Shen,et al.  Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method , 2006, J. Comput. Phys..

[3]  Tom Ilmanen,et al.  Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature , 1993 .

[4]  Andrea L. Bertozzi,et al.  A Wavelet-Laplace Variational Technique for Image Deconvolution and Inpainting , 2008, IEEE Transactions on Image Processing.

[5]  Takeshi Ohtsuka,et al.  Motion of interfaces by an Allen-Cahn type equation with multiple-well potentials , 2008, Asymptot. Anal..

[6]  James A. Warren,et al.  An efficient algorithm for solving the phase field crystal model , 2008, J. Comput. Phys..

[7]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[8]  Junseok Kim,et al.  An unconditionally gradient stable numerical method for solving the Allen-Cahn equation , 2009 .

[9]  Phillip Colella,et al.  Adaptive mesh refinement for multiscale nonequilibrium physics , 2005, Computing in Science & Engineering.

[10]  Jeffrey W. Bullard,et al.  Computational and mathematical models of microstructural evolution , 1998 .

[11]  Wheeler,et al.  Phase-field model for isothermal phase transitions in binary alloys. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[12]  Haijun Wu,et al.  A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen–Cahn Equation and the Mean Curvature Flow , 2005, J. Sci. Comput..

[13]  D. Duffy.  Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach , 2006 .

[14]  Karol Mikula,et al.  Simulation of anisotropic motion by mean curvature -- comparison of phase field and sharp interface approaches. , 1998 .

[15]  B. Vollmayr-Lee,et al.  Fast and accurate coarsening simulation with an unconditionally stable time step. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  P. Souganidis,et al.  Phase Transitions and Generalized Motion by Mean Curvature , 1992 .

[17]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[18]  Vittorio Cristini,et al.  Three-dimensional crystal growth-II: nonlinear simulation and control of the Mullins-Sekerka instability , 2004 .

[19]  P. Fife.  Dynamics of Internal Layers and Diffusive Interfaces , 1988 .

[20]  Weizhu Bao.  Approximation and comparison for motion by mean curvature with intersection points , 2003 .

[21]  E. Mello,et al.  Numerical study of the Cahn–Hilliard equation in one, two and three dimensions , 2004, cond-mat/0410772.

[22]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[23]  Craig Cowan,et al.  The Cahn-Hilliard Equation as a Gradient Flow , 2005 .

[24]  Zi-Kui Liu,et al.  Spectral implementation of an adaptive moving mesh method for phase-field equations , 2006, J. Comput. Phys..

[25]  G Grinstein,et al.  Directions in condensed matter physics : memorial volume in honor of Shang-keng Ma , 1986 .

[26]  F. Reitich,et al.  Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions , 1995 .

[27]  K. Mikula,et al.  Geometrical image segmentation by the Allen-Cahn equation , 2004 .

[28]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[29]  Qiang Du,et al.  STABILITY ANALYSIS AND APPLICATION OF THE EXPONENTIAL TIME DIFFERENCING SCHEMES 1) , 2004 .

[30]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[31]  Mowei Cheng,et al.  Maximally fast coarsening algorithms. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  J. Miller.  Numerical Analysis , 1966, Nature.