TY - JOUR
T1 - Verification of Convergence Rates of Numerical Solutions for Parabolic Equations
AU - Jeong, Darae
AU - Li, Yibao
AU - Lee, Chaeyoung
AU - Yang, Junxiang
AU - Choi, Yongho
AU - Kim, Junseok
N1 - Funding Information:
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 Korea University Future Research Grant. The authors (C. Lee and J. Yang) express their thanks for the support from the BK21 PLUS program.
Publisher Copyright:
© 2019 Darae Jeong et al.
PY - 2019
Y1 - 2019
N2 - In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen-Cahn equation, and the Cahn-Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.
AB - In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen-Cahn equation, and the Cahn-Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.
UR - http://www.scopus.com/inward/record.url?scp=85069042218&partnerID=8YFLogxK
U2 - 10.1155/2019/8152136
DO - 10.1155/2019/8152136
M3 - Article
AN - SCOPUS:85069042218
SN - 1024-123X
VL - 2019
JO - Mathematical Problems in Engineering
JF - Mathematical Problems in Engineering
M1 - 8152136
ER -