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 |
Bibliographical note
Funding Information:The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) ( NRF-2014R1A2A2A01003683 ). The authors greatly appreciate the reviewers for their constructive comments and suggestions, which have improved the quality of this paper.
Publisher Copyright:
© 2016 Elsevier Inc.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
Keywords
- Cahn–Hilliard equation
- Finite difference method
- Multigrid method
- Nonuniform grid
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics