Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Darae Jeong, Yibao Li, Chaeyoung Lee, Junxiang Yang, Yongho Choi, Junseok Kim

Research output: Contribution to journalArticle

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.

Original languageEnglish
Article number8152136
JournalMathematical Problems in Engineering
Volume2019
DOIs
Publication statusPublished - 2019 Jan 1

Fingerprint

Parabolic Equation
Rate of Convergence
Numerical Solution
Convergence Rate
First-order
Allen-Cahn Equation
Cahn-Hilliard Equation
Heat Equation
Hot Temperature

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

Cite this

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations. / Jeong, Darae; Li, Yibao; Lee, Chaeyoung; Yang, Junxiang; Choi, Yongho; Kim, Junseok.

In: Mathematical Problems in Engineering, Vol. 2019, 8152136, 01.01.2019.

Research output: Contribution to journalArticle

Jeong, Darae ; Li, Yibao ; Lee, Chaeyoung ; Yang, Junxiang ; Choi, Yongho ; Kim, Junseok. / Verification of Convergence Rates of Numerical Solutions for Parabolic Equations. In: Mathematical Problems in Engineering. 2019 ; Vol. 2019.
@article{3c5aca5fdec948e1af3caf8857f99957,
title = "Verification of Convergence Rates of Numerical Solutions for Parabolic Equations",
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.",
author = "Darae Jeong and Yibao Li and Chaeyoung Lee and Junxiang Yang and Yongho Choi and Junseok Kim",
year = "2019",
month = "1",
day = "1",
doi = "10.1155/2019/8152136",
language = "English",
volume = "2019",
journal = "Mathematical Problems in Engineering",
issn = "1024-123X",
publisher = "Hindawi Publishing Corporation",

}

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

PY - 2019/1/1

Y1 - 2019/1/1

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

UR - http://www.scopus.com/inward/citedby.url?scp=85069042218&partnerID=8YFLogxK

U2 - 10.1155/2019/8152136

DO - 10.1155/2019/8152136

M3 - Article

AN - SCOPUS:85069042218

VL - 2019

JO - Mathematical Problems in Engineering

JF - Mathematical Problems in Engineering

SN - 1024-123X

M1 - 8152136

ER -