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Title: Domain decomposition-based exponential time differencing methods for semilinear parabolic equations

Abstract

The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen-Cahn equation as a special case. We initially study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to confirm the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space.

Authors:
 [1];  [2];  [3]
+ Show Author Affiliations
  1. Hong Kong Polytechnic Univ. (Hong Kong)
  2. Univ. of South Carolina, Columbia, SC (United States)
  3. Auburn Univ., AL (United States)
Publication Date:
Research Org.:
Univ. of South Carolina, Columbia, SC (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Biological and Environmental Research (BER). Climate and Environmental Sciences Division; National Natural Science Foundation of China (NSFC); National Science Foundation (NSF)
OSTI Identifier:
1631280
Grant/Contract Number:  
SC0020270; DMS-1818438; SC0016540
Resource Type:
Accepted Manuscript
Journal Name:
BIT Numerical Mathematics
Additional Journal Information:
Journal Volume: 61; Journal ID: ISSN 0006-3835
Publisher:
Springer Nature
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; semilinear parabolic equation; overlapping domain decompositio; localized exponential time differencing; parallel Schwarz iteration

Citation Formats

Li, Xiao, Ju, Lili, and Hoang, Thi-Thao-Phuong. Domain decomposition-based exponential time differencing methods for semilinear parabolic equations. United States: N. p., 2020. Web. doi:10.1007/s10543-020-00817-0.
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Li, Xiao, Ju, Lili, & Hoang, Thi-Thao-Phuong. Domain decomposition-based exponential time differencing methods for semilinear parabolic equations. United States. https://doi.org/10.1007/s10543-020-00817-0
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Li, Xiao, Ju, Lili, and Hoang, Thi-Thao-Phuong. Fri . "Domain decomposition-based exponential time differencing methods for semilinear parabolic equations". United States. https://doi.org/10.1007/s10543-020-00817-0. https://www.osti.gov/servlets/purl/1631280.
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@article{osti_1631280,
title = {Domain decomposition-based exponential time differencing methods for semilinear parabolic equations},
author = {Li, Xiao and Ju, Lili and Hoang, Thi-Thao-Phuong},
abstractNote = {The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen-Cahn equation as a special case. We initially study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to confirm the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space.},
doi = {10.1007/s10543-020-00817-0},
journal = {BIT Numerical Mathematics},
number = ,
volume = 61,
place = {United States},
year = {Fri May 01 00:00:00 EDT 2020},
month = {Fri May 01 00:00:00 EDT 2020}
}
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https://doi.org/10.1007/s10543-020-00817-0
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