A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation

Yibao Li,Jian Wang,Jintae Park,Chaeyoung Lee,Sungha Yoon,Junseok Kim,Manuel De La Sen

doi:10.1155/2021/8889603

Abstract

We present a very simple benchmark problem for the numerical methods of the Cahn–Hilliard (CH) equation. For the benchmark problem, we consider a cosine function as the initial condition. The periodic sinusoidal profile satisfies both the homogeneous and periodic boundary conditions. The strength of the proposed problem is that it is simpler than the previous works. For the benchmark numerical solution of the CH equation, we use a fourth-order Runge–Kutta method (RK4) for the temporal integration and a centered finite difference scheme for the spatial differential operator. Using the proposed benchmark problem solution, we perform the convergence tests for an unconditionally gradient stable scheme via linear convex splitting proposed by Eyre and the Crank–Nicolson scheme. We obtain the expected convergence rates in time for the numerical schemes for the one-, two-, and three-dimensional CH equations.

Highlights

We present a very simple benchmark problem for the numerical methods of the following Cahn–Hilliard (CH) equation [1,2,3]: zφ(x, t) Δ􏽨F′(φ(x, t)) − ε2Δφ(x, t)􏽩, (x, t) ∈ Ω ×(0, T], zt (1)
Where Ω ⊂ Rd(d 1, 2, 3) is a bounded domain, φ(x, t) is a compositional field, F(φ) 0.25(φ2 − 1)2, and ε is a positive constant. e CH equation was proposed for a model of the spinodal decomposition in a binary mixture and has been used to model many scientific phenomena such as topology optimization [4], phase separation [5,6,7], image processing [8], two-phase fluid flows [9, 10], crystal model [11], tumor growth [12, 13], and microstructure formations
Erefore, the main purpose of this paper is to present a very simple benchmark problem for the numerical methods of the CH equation, which does not employ the self-test but a classical explicit method for the temporal discretization. e strength of the proposed problem is that it is simpler than the previous works [20,21,22]

Summary

Introduction

We present a very simple benchmark problem for the numerical methods of the following Cahn–Hilliard (CH) equation [1,2,3]: zφ(x, t) Δ􏽨F′(φ(x, t)) − ε2Δφ(x, t)􏽩, (x, t) ∈ Ω ×(0, T], zt (1). There are only few benchmark problems for validating the accuracy of the proposed numerical methods. The authors in [21] proposed a benchmark problem for the two- and threedimensional CH equations. E authors in [22] presented four benchmark problems for the Allen–Cahn (AC) and CH equations. E benchmark is the time T at which the value at a point in the domain changes from negative to positive. E authors in [20] presented two benchmark problems for phase-field models of solute diffusion and phase separation.

Objectives

Methods

Results

Conclusion