A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation. (9th March 2021)

• Record Type: Journal Article
• Authors: Li, Yibao; Lee, Chaeyoung; Wang, Jian; Yoon, Sungha; Park, Jintae; Kim, Junseok
• Other Names: De la Sen Manuel Academic Editor.
• Is Part Of: Discrete dynamics in nature and society. Volume 2021(2021)
• Publication Date: 2021-03-09
• Subjects: System analysis -- Periodicals; Dynamics -- Periodicals; Chaotic behavior in systems -- Periodicals; Differentiable dynamical systems -- Periodicals; 003.05
• DOI: 10.1155/2021/8889603
• Languages: English
• ISSNs: 1026-0226

Abstract

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.