skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

This content will become publicly available on January 29, 2021

Title: Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems

Abstract

In this paper, we propose nonoverlapping localized exponential time differencing (ETD) methods for diffusion problems. The model time-dependent diffusion equation is first reformulated on subdomains based on the nonoverlapping domain decomposition, in which Neumann boundary conditions are imposed on the interfaces for the subdomain problems and Dirichlet type conditions are enforced to form a space-time interface problem. After spatial discretization by standard central finite differences and temporal integration with the first or second order ETD methods, the fully discrete interface problem is obtained. Such an interface problem is then solved iteratively either at each time step or over the whole time interval: the former involves the solution of stationary problems in each subdomain at each iteration while the latter involves the solution of time-dependent subdomain problems at each iteration. For both approaches, we prove that localized ETD solutions conserve mass exactly and converge in time to the exact space semidiscrete solution. Numerical experiments in two dimensions are also presented to illustrate the performance of the proposed methods.

Authors:
 [1];  [2];  [3];  [2]
  1. Auburn Univ., AL (United States)
  2. Univ. of South Carolina, Columbia, SC (United States)
  3. Chinese Academy of Sciences (CAS), Beijing (China). State Key Laboratory of Scientific and Engineering Computing
Publication Date:
Research Org.:
Univ. of South Carolina, Columbia, SC (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Biological and Environmental Research (BER) (SC-23). Climate and Environmental Sciences Division; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); National Science Foundation (NSF)
OSTI Identifier:
1594011
Grant/Contract Number:  
SC0016540; SC0020270; DMS-1818438; DMS-1913073
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Scientific Computing
Additional Journal Information:
Journal Volume: 82; Journal Issue: 2; Journal ID: ISSN 0885-7474
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING; Exponential time differencing; Nonoverlapping domain decomposition; Mass conservation; Finite differences; Cross points

Citation Formats

Hoang, Thi-Thao-Phuong, Ju, Lili, Leng, Wei, and Wang, Zhu. Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems. United States: N. p., 2020. Web. doi:10.1007/s10915-020-01136-w.
Hoang, Thi-Thao-Phuong, Ju, Lili, Leng, Wei, & Wang, Zhu. Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems. United States. doi:10.1007/s10915-020-01136-w.
Hoang, Thi-Thao-Phuong, Ju, Lili, Leng, Wei, and Wang, Zhu. Wed . "Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems". United States. doi:10.1007/s10915-020-01136-w.
@article{osti_1594011,
title = {Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems},
author = {Hoang, Thi-Thao-Phuong and Ju, Lili and Leng, Wei and Wang, Zhu},
abstractNote = {In this paper, we propose nonoverlapping localized exponential time differencing (ETD) methods for diffusion problems. The model time-dependent diffusion equation is first reformulated on subdomains based on the nonoverlapping domain decomposition, in which Neumann boundary conditions are imposed on the interfaces for the subdomain problems and Dirichlet type conditions are enforced to form a space-time interface problem. After spatial discretization by standard central finite differences and temporal integration with the first or second order ETD methods, the fully discrete interface problem is obtained. Such an interface problem is then solved iteratively either at each time step or over the whole time interval: the former involves the solution of stationary problems in each subdomain at each iteration while the latter involves the solution of time-dependent subdomain problems at each iteration. For both approaches, we prove that localized ETD solutions conserve mass exactly and converge in time to the exact space semidiscrete solution. Numerical experiments in two dimensions are also presented to illustrate the performance of the proposed methods.},
doi = {10.1007/s10915-020-01136-w},
journal = {Journal of Scientific Computing},
number = 2,
volume = 82,
place = {United States},
year = {2020},
month = {1}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on January 29, 2021
Publisher's Version of Record

Save / Share:

Works referenced in this record:

A homographic best approximation problem with application to optimized Schwarz waveform relaxation
journal, January 2009


Exponential Time Differencing for Stiff Systems
journal, March 2002


Analysis of inexact Krylov subspace methods for approximating the matrix exponential
journal, August 2017


Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications
journal, June 2005


Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen--Cahn Equation
journal, January 2019

  • Du, Qiang; Ju, Lili; Li, Xiao
  • SIAM Journal on Numerical Analysis, Vol. 57, Issue 2
  • DOI: 10.1137/18M118236X

A scalable Lagrange multiplier based domain decomposition method for time-dependent problems
journal, November 1995

  • Farhat, Charbel; Chen, Po-Shu; Mandel, Jan
  • International Journal for Numerical Methods in Engineering, Vol. 38, Issue 22
  • DOI: 10.1002/nme.1620382207

Optimal convergence properties of the FETI domain decomposition method
journal, May 1994

  • Farhat, Charbel; Mandel, Jan; Roux, Francois Xavier
  • Computer Methods in Applied Mechanics and Engineering, Vol. 115, Issue 3-4
  • DOI: 10.1016/0045-7825(94)90068-X

A method of finite element tearing and interconnecting and its parallel solution algorithm
journal, October 1991

  • Farhat, Charbel; Roux, Francois-Xavier
  • International Journal for Numerical Methods in Engineering, Vol. 32, Issue 6
  • DOI: 10.1002/nme.1620320604

Efficient Solution of Parabolic Equations by Krylov Approximation Methods
journal, September 1992

  • Gallopoulos, E.; Saad, Y.
  • SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 5
  • DOI: 10.1137/0913071

Optimized Schwarz Methods
journal, January 2006


An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere
journal, October 2016


Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations
journal, January 2013

  • Hoang, Thi-Thao-Phuong; Jaffré, Jérôme; Japhet, Caroline
  • SIAM Journal on Numerical Analysis, Vol. 51, Issue 6
  • DOI: 10.1137/130914401

Overlapping localized exponential time differencing methods for diffusion problems
journal, January 2018

  • Hoang, Thi-Thao-Phuong; Ju, Lili; Wang, Zhu
  • Communications in Mathematical Sciences, Vol. 16, Issue 6
  • DOI: 10.4310/CMS.2018.v16.n6.a3

On Krylov Subspace Approximations to the Matrix Exponential Operator
journal, October 1997


Exponential Integrators for Large Systems of Differential Equations
journal, September 1998

  • Hochbruck, Marlis; Lubich, Christian; Selhofer, Hubert
  • SIAM Journal on Scientific Computing, Vol. 19, Issue 5
  • DOI: 10.1137/S1064827595295337

Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems
journal, January 2005

  • Hochbruck, Marlis; Ostermann, Alexander
  • SIAM Journal on Numerical Analysis, Vol. 43, Issue 3
  • DOI: 10.1137/040611434

Exponential integrators
journal, May 2010


Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection–diffusion–reaction equations
journal, November 2013


Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations
journal, May 2014


An ETD Crank-Nicolson method for reaction-diffusion systems
journal, May 2011

  • Kleefeld, B.; Khaliq, A. Q. M.; Wade, B. A.
  • Numerical Methods for Partial Differential Equations, Vol. 28, Issue 4
  • DOI: 10.1002/num.20682

Generalized integrating factor methods for stiff PDEs
journal, February 2005


Characteristic exponents and diagonally dominant linear differential systems
journal, July 1971


Implementation of Parallel Adaptive-Krylov Exponential Solvers for Stiff Problems
journal, January 2014

  • Loffeld, J.; Tokman, M.
  • SIAM Journal on Scientific Computing, Vol. 36, Issue 5
  • DOI: 10.1137/13094462X

Compact integration factor methods in high spatial dimensions
journal, May 2008

  • Nie, Qing; Wan, Frederic Y. M.; Zhang, Yong-Tao
  • Journal of Computational Physics, Vol. 227, Issue 10
  • DOI: 10.1016/j.jcp.2008.01.050

Algorithm 919: A Krylov Subspace Algorithm for Evaluating the ϕ-Functions Appearing in Exponential Integrators
journal, April 2012

  • Niesen, Jitse; Wright, Will M.
  • ACM Transactions on Mathematical Software, Vol. 38, Issue 3
  • DOI: 10.1145/2168773.2168781

Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator
journal, February 1992

  • Saad, Y.
  • SIAM Journal on Numerical Analysis, Vol. 29, Issue 1
  • DOI: 10.1137/0729014

Expokit: a software package for computing matrix exponentials
journal, March 1998


Fourth-Order Time-Stepping for Stiff PDEs
journal, January 2005

  • Kassam, Aly-Khan; Trefethen, Lloyd N.
  • SIAM Journal on Scientific Computing, Vol. 26, Issue 4
  • DOI: 10.1137/S1064827502410633

Error Bounds for the Lanczos Methods for Approximating Matrix Exponentials
journal, January 2013

  • Ye, Qiang
  • SIAM Journal on Numerical Analysis, Vol. 51, Issue 1
  • DOI: 10.1137/11085935X

Extreme-Scale Phase Field Simulations of Coarsening Dynamics on the Sunway TaihuLight Supercomputer
conference, November 2016

  • Zhang, Jian; Zhou, Chunbao; Wang, Yangang
  • SC16: International Conference for High Performance Computing, Networking, Storage and Analysis
  • DOI: 10.1109/SC.2016.3