Volume 26, Issue 5
Convergence Analysis of Exponential Time Differencing Schemes for the Cahn-Hilliard Equation

Commun. Comput. Phys., 26 (2019), pp. 1510-1529.

Published online: 2019-08

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• Abstract

In this paper, we rigorously prove the convergence of fully discrete first- and second-order exponential time differencing schemes for solving the Cahn-Hilliard equation. Our analyses mainly follow the standard procedure with the consistency and stability estimates for numerical error functions, while the technique of higher-order consistency analysis is adopted in order to obtain the uniform L boundedness of the numerical solutions under some moderate constraints on the time step and spatial mesh sizes. This paper provides a theoretical support for numerical analysis of exponential time differencing and other related numerical methods for phase field models, in which an assumption on the uniform L boundedness is usually needed.

• Keywords

Cahn-Hilliard equation, exponential time differencing, convergence analysis, uniform $L^∞$ boundedness.

35K55, 65M12, 65M15, 65F30

li434@mailbox.sc.edu (Xiao Li)

ju@math.sc.edu (Lili Ju)

mengx@mailbox.sc.edu (Xucheng Meng)

• BibTex
• RIS
• TXT
@Article{CiCP-26-1510, author = {Li , Xiao and Ju , Lili and Meng , Xucheng}, title = {Convergence Analysis of Exponential Time Differencing Schemes for the Cahn-Hilliard Equation}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {5}, pages = {1510--1529}, abstract = {

In this paper, we rigorously prove the convergence of fully discrete first- and second-order exponential time differencing schemes for solving the Cahn-Hilliard equation. Our analyses mainly follow the standard procedure with the consistency and stability estimates for numerical error functions, while the technique of higher-order consistency analysis is adopted in order to obtain the uniform L boundedness of the numerical solutions under some moderate constraints on the time step and spatial mesh sizes. This paper provides a theoretical support for numerical analysis of exponential time differencing and other related numerical methods for phase field models, in which an assumption on the uniform L boundedness is usually needed.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2019.js60.12}, url = {http://global-sci.org/intro/article_detail/cicp/13274.html} }
TY - JOUR T1 - Convergence Analysis of Exponential Time Differencing Schemes for the Cahn-Hilliard Equation AU - Li , Xiao AU - Ju , Lili AU - Meng , Xucheng JO - Communications in Computational Physics VL - 5 SP - 1510 EP - 1529 PY - 2019 DA - 2019/08 SN - 26 DO - http://doi.org/10.4208/cicp.2019.js60.12 UR - https://global-sci.org/intro/article_detail/cicp/13274.html KW - Cahn-Hilliard equation, exponential time differencing, convergence analysis, uniform $L^∞$ boundedness. AB -

In this paper, we rigorously prove the convergence of fully discrete first- and second-order exponential time differencing schemes for solving the Cahn-Hilliard equation. Our analyses mainly follow the standard procedure with the consistency and stability estimates for numerical error functions, while the technique of higher-order consistency analysis is adopted in order to obtain the uniform L boundedness of the numerical solutions under some moderate constraints on the time step and spatial mesh sizes. This paper provides a theoretical support for numerical analysis of exponential time differencing and other related numerical methods for phase field models, in which an assumption on the uniform L boundedness is usually needed.

Xiao Li, Lili Ju & Xucheng Meng. (2019). Convergence Analysis of Exponential Time Differencing Schemes for the Cahn-Hilliard Equation. Communications in Computational Physics. 26 (5). 1510-1529. doi:10.4208/cicp.2019.js60.12
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