### 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 |
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Pages (from-to) | 320-333 |

Number of pages | 14 |

Journal | Applied Mathematics and Computation |

Volume | 293 |

DOIs | |

Publication status | Published - 2017 Jan 15 |

### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

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## Cite this

*Applied Mathematics and Computation*,

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