TY - JOUR
T1 - A multigrid solution for the Cahn–Hilliard equation on nonuniform grids
AU - Choi, Yongho
AU - Jeong, Darae
AU - Kim, Junseok
N1 - 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.
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
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U2 - 10.1016/j.amc.2016.08.026
DO - 10.1016/j.amc.2016.08.026
M3 - Article
AN - SCOPUS:84985906573
VL - 293
SP - 320
EP - 333
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
SN - 0096-3003
ER -