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.
Highlights
We summarize here the nonlinear multigrid method for solving the discrete CH system as follows: First, let us rewrite Equations (5) and (6) as NSO(φn+1, μn+1 ) = (ξ n, ψn ), where the linear operator NSO is defined as NSO(φn+1, μn+1 ) = φn+1 /∆t − ∆h μn+1, μn+1 −3 + e2 ∆h φn+1, and the source term is denoted by (ξ n, ψn ) =
We presented a nonlinear multigrid implementation for the CH equation in a two-dimensional space
We described the implementation of our numerical scheme in detail
Summary
We consider a detailed multigrid [1] implementation of the following two-dimensional Cahn–Hilliard (CH) equation [2] and provide its C source code:. The second boundary condition (2) implies that the total mass is conserved. Conservation of mass implies the following CH equation φt = −∇ · F , where the flux is given by F = − M ∇μ. Details regarding the implementation, multigrid performance, and source codes have not been provided. The main purpose of this paper is to describe a detailed multigrid implementation of the two-dimensional CH equation, evaluate its performance and provide its C programming language source code. In the Appendix A, we provide the C code for implementing the nonlinear multigrid technique for the two-dimensional CH equation
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