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Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Authors
Jeong, DaraeLi, YibaoLee, ChaeyoungYang, JunxiangChoi, YonghoKim, Junseok
Issue Date
2019
Publisher
HINDAWI LTD
Citation
MATHEMATICAL PROBLEMS IN ENGINEERING, v.2019
Indexed
SCIE
SCOPUS
Journal Title
MATHEMATICAL PROBLEMS IN ENGINEERING
Volume
2019
URI
https://scholar.korea.ac.kr/handle/2021.sw.korea/131723
DOI
10.1155/2019/8152136
ISSN
1024-123X
Abstract
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.
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