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

作者：Choi Yongho; Jeong Darae; Kim Junseok^*

来源：Applied Mathematics and Computation, 2017, 293: 320-333.

DOI：10.1016/j.amc.2016.08.026

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

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.

• 出版日期2017-1-15