### Abstract

We present a nonlinear multigrid method to solve the Cahn–Hilliard (CH) equation on nonuniform grids. The CH equation was originally proposed as a mathematical model to describe phase separation phenomena after the quenching of binary alloys. The model has the characteristics of thin diffusive interfaces. To resolve the sharp interfacial transition, we need a very fine grid, which is computationally expensive. To reduce the cost, we can use a fine grid around the interfacial transition region and a relatively coarser grid in the bulk region. The CH equation is discretized by a conservative finite difference scheme in space and an unconditionally gradient stable type scheme in time. We use a conservative restriction in the nonlinear multigrid method to conserve the total mass in the coarser grid levels. Various numerical results on one-, two-, and three-dimensional spaces are presented to demonstrate the accuracy and effectiveness of the nonuniform grids for the CH equation.

Original language | English |
---|---|

Pages (from-to) | 320-333 |

Number of pages | 14 |

Journal | Applied Mathematics and Computation |

Volume | 293 |

DOIs | |

Publication status | Published - 2017 Jan 15 |

### Fingerprint

### Keywords

- Cahn–Hilliard equation
- Finite difference method
- Multigrid method
- Nonuniform grid

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Applied Mathematics and Computation*,

*293*, 320-333. https://doi.org/10.1016/j.amc.2016.08.026

**A multigrid solution for the Cahn–Hilliard equation on nonuniform grids.** / Choi, Yongho; Jeong, Darae; Kim, Junseok.

Research output: Contribution to journal › Article

*Applied Mathematics and Computation*, vol. 293, pp. 320-333. https://doi.org/10.1016/j.amc.2016.08.026

}

TY - JOUR

T1 - A multigrid solution for the Cahn–Hilliard equation on nonuniform grids

AU - Choi, Yongho

AU - Jeong, Darae

AU - Kim, Junseok

PY - 2017/1/15

Y1 - 2017/1/15

N2 - We present a nonlinear multigrid method to solve the Cahn–Hilliard (CH) equation on nonuniform grids. The CH equation was originally proposed as a mathematical model to describe phase separation phenomena after the quenching of binary alloys. The model has the characteristics of thin diffusive interfaces. To resolve the sharp interfacial transition, we need a very fine grid, which is computationally expensive. To reduce the cost, we can use a fine grid around the interfacial transition region and a relatively coarser grid in the bulk region. The CH equation is discretized by a conservative finite difference scheme in space and an unconditionally gradient stable type scheme in time. We use a conservative restriction in the nonlinear multigrid method to conserve the total mass in the coarser grid levels. Various numerical results on one-, two-, and three-dimensional spaces are presented to demonstrate the accuracy and effectiveness of the nonuniform grids for the CH equation.

AB - We present a nonlinear multigrid method to solve the Cahn–Hilliard (CH) equation on nonuniform grids. The CH equation was originally proposed as a mathematical model to describe phase separation phenomena after the quenching of binary alloys. The model has the characteristics of thin diffusive interfaces. To resolve the sharp interfacial transition, we need a very fine grid, which is computationally expensive. To reduce the cost, we can use a fine grid around the interfacial transition region and a relatively coarser grid in the bulk region. The CH equation is discretized by a conservative finite difference scheme in space and an unconditionally gradient stable type scheme in time. We use a conservative restriction in the nonlinear multigrid method to conserve the total mass in the coarser grid levels. Various numerical results on one-, two-, and three-dimensional spaces are presented to demonstrate the accuracy and effectiveness of the nonuniform grids for the CH equation.

KW - Cahn–Hilliard equation

KW - Finite difference method

KW - Multigrid method

KW - Nonuniform grid

UR - http://www.scopus.com/inward/record.url?scp=84985906573&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84985906573&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2016.08.026

DO - 10.1016/j.amc.2016.08.026

M3 - Article

VL - 293

SP - 320

EP - 333

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

ER -