Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation

Mikhail A. Bochev, I.V. Oseledets, E.E. Tyrtyshnikov

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)
22 Downloads (Pure)

Abstract

The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation (WR) method based on block Krylov subspaces. Second, we compare this new WR-Krylov implementation against Krylov subspace methods combined with the shift and invert (SAI) technique. Some analysis and numerical experiments are presented. Since the WR-Krylov and SAI-Krylov methods build up the solution simultaneously for the whole time interval and there is no time stepping involved, both methods can be seen as iterative across-time methods. The key difference between these methods and standard time integration methods is that their accuracy is not directly related to the time step size.
Original languageEnglish
Pages (from-to)2088-2098
Number of pages11
JournalComputers and mathematics with applications
Volume67
Issue number12
DOIs
Publication statusPublished - Jul 2014

Fingerprint

Waveform Relaxation
Krylov Subspace
Linear differential equation
Differential equations
Invert
Experiments
Waveform Relaxation Method
Krylov Methods
Krylov Subspace Methods
Time Stepping
Time Integration
Efficient Implementation
Continuous Time
Fold
Numerical Experiment
Interval

Keywords

  • EWI-25568
  • MSC-65F60
  • MSC-65L05
  • MSC-65N22
  • MSC-65F10
  • MSC-65F30
  • Waveform relaxation
  • Residual
  • Richardson iteration
  • Low rank approximation
  • Krylov subspace methods
  • METIS-309814
  • IR-93648
  • Matrix exponential
  • Anderson acceleration

Cite this

Bochev, Mikhail A. ; Oseledets, I.V. ; Tyrtyshnikov, E.E. / Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. In: Computers and mathematics with applications. 2014 ; Vol. 67, No. 12. pp. 2088-2098.
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abstract = "The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation (WR) method based on block Krylov subspaces. Second, we compare this new WR-Krylov implementation against Krylov subspace methods combined with the shift and invert (SAI) technique. Some analysis and numerical experiments are presented. Since the WR-Krylov and SAI-Krylov methods build up the solution simultaneously for the whole time interval and there is no time stepping involved, both methods can be seen as iterative across-time methods. The key difference between these methods and standard time integration methods is that their accuracy is not directly related to the time step size.",
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Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. / Bochev, Mikhail A.; Oseledets, I.V.; Tyrtyshnikov, E.E.

In: Computers and mathematics with applications, Vol. 67, No. 12, 07.2014, p. 2088-2098.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation

AU - Bochev, Mikhail A.

AU - Oseledets, I.V.

AU - Tyrtyshnikov, E.E.

N1 - The last name of the first author can also be spelled as "Bochev".

PY - 2014/7

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N2 - The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation (WR) method based on block Krylov subspaces. Second, we compare this new WR-Krylov implementation against Krylov subspace methods combined with the shift and invert (SAI) technique. Some analysis and numerical experiments are presented. Since the WR-Krylov and SAI-Krylov methods build up the solution simultaneously for the whole time interval and there is no time stepping involved, both methods can be seen as iterative across-time methods. The key difference between these methods and standard time integration methods is that their accuracy is not directly related to the time step size.

AB - The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation (WR) method based on block Krylov subspaces. Second, we compare this new WR-Krylov implementation against Krylov subspace methods combined with the shift and invert (SAI) technique. Some analysis and numerical experiments are presented. Since the WR-Krylov and SAI-Krylov methods build up the solution simultaneously for the whole time interval and there is no time stepping involved, both methods can be seen as iterative across-time methods. The key difference between these methods and standard time integration methods is that their accuracy is not directly related to the time step size.

KW - EWI-25568

KW - MSC-65F60

KW - MSC-65L05

KW - MSC-65N22

KW - MSC-65F10

KW - MSC-65F30

KW - Waveform relaxation

KW - Residual

KW - Richardson iteration

KW - Low rank approximation

KW - Krylov subspace methods

KW - METIS-309814

KW - IR-93648

KW - Matrix exponential

KW - Anderson acceleration

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DO - 10.1016/j.camwa.2014.03.002

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