TY - JOUR
T1 - A benchmark problem for the two- and three-dimensional Cahn–Hilliard equations
AU - Jeong, Darae
AU - Choi, Yongho
AU - Kim, Junseok
N1 - 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.
PY - 2018/8
Y1 - 2018/8
N2 - 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.
AB - 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.
KW - Benchmark problem
KW - Cahn–Hilliard equation
KW - Finite difference method
KW - Multigrid method
UR - http://www.scopus.com/inward/record.url?scp=85042187778&partnerID=8YFLogxK
U2 - 10.1016/j.cnsns.2018.02.006
DO - 10.1016/j.cnsns.2018.02.006
M3 - Article
AN - SCOPUS:85042187778
SN - 1007-5704
VL - 61
SP - 149
EP - 159
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
ER -