Abstract
This paper proposes a benchmark problem for the two- and three-dimensional Cahn–Hilliard (CH) equations, which describe the process of phase separation. The CH equation is highly nonlinear and an analytical solution does not exist except trivial solutions. Therefore, we have to approximate the CH equation numerically. To test the accuracy of a numerical scheme, we have to resort to convergence tests, which consist of consecutive relative errors or a very fine solution from the numerical scheme. For a fair convergence test, we provide benchmark problems which are of the shrinking annulus and spherical shell type. We show numerical results by using the explicit Euler's scheme with a very fine time step size and also present a comparison test with Eyre's convex splitting schemes.
| Original language | English |
|---|---|
| Pages (from-to) | 149-159 |
| Number of pages | 11 |
| Journal | Communications in Nonlinear Science and Numerical Simulation |
| Volume | 61 |
| DOIs | |
| Publication status | Published - 2018 Aug |
Bibliographical note
Funding Information:The authors thank the reviewers for the constructive and helpful comments on the revision of this article. The first author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MSIP ) ( NRF-2017R1E1A1A03070953 ). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education ( NRF-2016R1D1A1B03933243 ).
Publisher Copyright:
© 2018 Elsevier B.V.
Keywords
- Benchmark problem
- Cahn–Hilliard equation
- Finite difference method
- Multigrid method
ASJC Scopus subject areas
- Numerical Analysis
- Modelling and Simulation
- Applied Mathematics