# A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D

## Abstract

We present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional Cahn–Hilliard–Cook (CHC) equation, which is a fourth-order stochastic partial differential equation with a noise term. The equation is discretized in space and time based on a fully implicit, cell-centered finite difference scheme, with an adaptive time-stepping strategy designed to accelerate the progress to equilibrium. At each time step, a parallel Newton–Krylov–Schwarz algorithm is used to solve the nonlinear system. We discuss various numerical and computational challenges associated with the method. The numerical scheme is validated by a comparison with an explicit scheme of high accuracy (and unreasonably high cost). We present steady state solutions of the CHC equation in two and three dimensions. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise. We also show the evolution of the energies and statistical moments. In terms of the parallel performance, it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors.

- Authors:

- Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027 (United States)
- Institute of Software, Chinese Academy of Sciences, Beijing 100190 (China)
- (China)
- Department of Computer Science, University of Colorado Boulder, Boulder, CO 80309 (United States)
- Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal 23955 (Saudi Arabia)
- (United States)

- Publication Date:

- OSTI Identifier:
- 22465608

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 285; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ACCURACY; ALGORITHMS; FIELD EQUATIONS; FLUCTUATIONS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; PARTIAL DIFFERENTIAL EQUATIONS; STEADY-STATE CONDITIONS; STOCHASTIC PROCESSES; SUPERCOMPUTERS; THREE-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Zheng, Xiang, Yang, Chao, State Key Laboratory of Computer Science, Chinese Academy of Sciences, Beijing 100190, Cai, Xiao-Chuan, E-mail: cai@cs.colorado.edu, Keyes, David, and Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027.
```*A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2015.01.016.

```
Zheng, Xiang, Yang, Chao, State Key Laboratory of Computer Science, Chinese Academy of Sciences, Beijing 100190, Cai, Xiao-Chuan, E-mail: cai@cs.colorado.edu, Keyes, David, & Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027.
```*A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D*. United States. doi:10.1016/J.JCP.2015.01.016.

```
Zheng, Xiang, Yang, Chao, State Key Laboratory of Computer Science, Chinese Academy of Sciences, Beijing 100190, Cai, Xiao-Chuan, E-mail: cai@cs.colorado.edu, Keyes, David, and Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027. Sun .
"A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D". United States.
doi:10.1016/J.JCP.2015.01.016.
```

```
@article{osti_22465608,
```

title = {A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D},

author = {Zheng, Xiang and Yang, Chao and State Key Laboratory of Computer Science, Chinese Academy of Sciences, Beijing 100190 and Cai, Xiao-Chuan, E-mail: cai@cs.colorado.edu and Keyes, David and Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027},

abstractNote = {We present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional Cahn–Hilliard–Cook (CHC) equation, which is a fourth-order stochastic partial differential equation with a noise term. The equation is discretized in space and time based on a fully implicit, cell-centered finite difference scheme, with an adaptive time-stepping strategy designed to accelerate the progress to equilibrium. At each time step, a parallel Newton–Krylov–Schwarz algorithm is used to solve the nonlinear system. We discuss various numerical and computational challenges associated with the method. The numerical scheme is validated by a comparison with an explicit scheme of high accuracy (and unreasonably high cost). We present steady state solutions of the CHC equation in two and three dimensions. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise. We also show the evolution of the energies and statistical moments. In terms of the parallel performance, it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors.},

doi = {10.1016/J.JCP.2015.01.016},

journal = {Journal of Computational Physics},

number = ,

volume = 285,

place = {United States},

year = {Sun Mar 15 00:00:00 EDT 2015},

month = {Sun Mar 15 00:00:00 EDT 2015}

}