Nonlinear Multigrid Implementation for the Two-Dimensional Cahn–Hilliard Equation

Chaeyoung Lee, Darae Jeong, Junxiang Yang and Junseok Kim
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Chaeyoung Lee: Department of Mathematics, Korea University, Seoul 02841, Korea
Darae Jeong: Department of Mathematics, Kangwon National University, Chuncheon-si 200-090, Korea
Junxiang Yang: Department of Mathematics, Korea University, Seoul 02841, Korea
Junseok Kim: Department of Mathematics, Korea University, Seoul 02841, Korea

Mathematics, 2020, vol. 8, issue 1, 1-23

Abstract: We present a nonlinear multigrid implementation for the two-dimensional Cahn–Hilliard (CH) equation and conduct detailed numerical tests to explore the performance of the multigrid method for the CH equation. The CH equation was originally developed by Cahn and Hilliard to model phase separation phenomena. The CH equation has been used to model many interface-related problems, such as the spinodal decomposition of a binary alloy mixture, inpainting of binary images, microphase separation of diblock copolymers, microstructures with elastic inhomogeneity, two-phase binary fluids, in silico tumor growth simulation and structural topology optimization. The CH equation is discretized by using Eyre’s unconditionally gradient stable scheme. The system of discrete equations is solved using an iterative method such as a nonlinear multigrid approach, which is one of the most efficient iterative methods for solving partial differential equations. Characteristic numerical experiments are conducted to demonstrate the efficiency and accuracy of the multigrid method for the CH equation. In the Appendix, we provide C code for implementing the nonlinear multigrid method for the two-dimensional CH equation.

Keywords: Cahn–Hilliard equation; multigrid method; unconditionally gradient stable scheme (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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