Overlapping localized exponential time differencing methods for diffusion problems

Hoang, Thi-Thao-Phuong; Ju, Lili; Wang, Zhu

doi:10.4310/CMS.2018.v16.n6.a3

Title: Overlapping localized exponential time differencing methods for diffusion problems

Abstract

The localized exponential time differencing (ETD) based on overlapping domain decomposition has been recently introduced for extreme-scale phase field simulations of coarsening dynamics, which displays excellent parallel scalability in supercomputers. This paper serves as the first step toward building a solid mathematical foundation for this approach. We study the overlapping localized ETD schemes for a model time-dependent diffusion equation discretized in space by the standard central difference. Two methods are proposed and analyzed for solving the fully discrete localized ETD systems: the first one is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second one is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergences of the associated iterative solutions to the corresponding fully discrete localized ETD solution and to the exact semidiscrete solution are rigorously proved. Finally, numerical experiments are also carried out to confirm theoretical results and to compare the performance of the two methods.

Authors:

Hoang, Thi-Thao-Phuong ^[1]; Ju, Lili ^[1]; Wang, Zhu ^[1]

Univ. of South Carolina, Columbia, SC (United States)

Publication Date:: Thu Feb 07 00:00:00 EST 2019

Research Org.:: Univ. of South Carolina, Columbia, SC (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)

OSTI Identifier:: 1593992

Grant/Contract Number:: SC0016540; DMS-1521965

Resource Type:: Accepted Manuscript

Journal Name:: Communications in Mathematical Sciences

Additional Journal Information:: Journal Volume: 16; Journal Issue: 6; Journal ID: ISSN 1539-6746

Publisher:: International Press

Country of Publication:: United States

Language:: English

Subject:: 58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING; Exponential time differencing; overlapping domain decomposition; diffusion equation; localization; parallel Schwarz iteration; waveform relaxation

Citation Formats


                    Hoang, Thi-Thao-Phuong, Ju, Lili, and Wang, Zhu. Overlapping localized exponential time differencing methods for diffusion problems.  United States: N. p., 2019. 
Web.  doi:10.4310/CMS.2018.v16.n6.a3.

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                    Hoang, Thi-Thao-Phuong, Ju, Lili, & Wang, Zhu. Overlapping localized exponential time differencing methods for diffusion problems.  United States.  https://doi.org/10.4310/CMS.2018.v16.n6.a3

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                    Hoang, Thi-Thao-Phuong, Ju, Lili, and Wang, Zhu. Thu .  
"Overlapping localized exponential time differencing methods for diffusion problems".  United States.  https://doi.org/10.4310/CMS.2018.v16.n6.a3.  https://www.osti.gov/servlets/purl/1593992.

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@article{osti_1593992,

  title        = {Overlapping localized exponential time differencing methods for diffusion problems},

  author       = {Hoang, Thi-Thao-Phuong and Ju, Lili and Wang, Zhu},

  abstractNote = {The localized exponential time differencing (ETD) based on overlapping domain decomposition has been recently introduced for extreme-scale phase field simulations of coarsening dynamics, which displays excellent parallel scalability in supercomputers. This paper serves as the first step toward building a solid mathematical foundation for this approach. We study the overlapping localized ETD schemes for a model time-dependent diffusion equation discretized in space by the standard central difference. Two methods are proposed and analyzed for solving the fully discrete localized ETD systems: the first one is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second one is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergences of the associated iterative solutions to the corresponding fully discrete localized ETD solution and to the exact semidiscrete solution are rigorously proved. Finally, numerical experiments are also carried out to confirm theoretical results and to compare the performance of the two methods.},

  doi          = {10.4310/CMS.2018.v16.n6.a3},

  journal      = {Communications in Mathematical Sciences},

  number       = 6,

  volume       = 16,

  place        = {United States},

  year         = {Thu Feb 07 00:00:00 EST 2019},

  month        = {Thu Feb 07 00:00:00 EST 2019}

}

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Works referencing / citing this record:

Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems
journal, February 2020

Hoang, Thi-Thao-Phuong; Ju, Lili; Wang, Zhu
Journal of Scientific Computing, Vol. 82, Issue 2
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