Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations

Zdzislaw Jackiewicz, B. Zubik-Kowal

Research output: Contribution to journalArticle

44 Citations (Scopus)

Abstract

We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated in a parallel computing environment by highly stable implicit methods. The effectiveness of this approach is illustrated by numerical experiments on the Hutchinson's equation. The boundedness of waveform relaxation iterations is proved for the Hutchinson's equation. This result is used in the proof of the superlinear convergence of the iterations.

Original languageEnglish (US)
Pages (from-to)433-443
Number of pages11
JournalApplied Numerical Mathematics
Volume56
Issue number3-4 SPEC. ISS.
DOIs
StatePublished - Mar 2006

Fingerprint

Waveform Relaxation Method
Parallel processing systems
Delay Differential Equations
Collocation
Partial differential equations
Differential equations
Partial differential equation
Waveform Relaxation
Iteration
Superlinear Convergence
Implicit Method
Spectral Methods
Parallel Computing
Collocation Method
Chebyshev
Nonlinear Differential Equations
Boundedness
Experiments
Numerical Experiment

Keywords

  • Nonlinear partial delay differential equations
  • Pseudospectral methods
  • Waveform relaxation iterations

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Modeling and Simulation

Cite this

Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. / Jackiewicz, Zdzislaw; Zubik-Kowal, B.

In: Applied Numerical Mathematics, Vol. 56, No. 3-4 SPEC. ISS., 03.2006, p. 433-443.

Research output: Contribution to journalArticle

@article{a810f08d51064aef92cdb647f4c388bd,
title = "Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations",
abstract = "We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated in a parallel computing environment by highly stable implicit methods. The effectiveness of this approach is illustrated by numerical experiments on the Hutchinson's equation. The boundedness of waveform relaxation iterations is proved for the Hutchinson's equation. This result is used in the proof of the superlinear convergence of the iterations.",
keywords = "Nonlinear partial delay differential equations, Pseudospectral methods, Waveform relaxation iterations",
author = "Zdzislaw Jackiewicz and B. Zubik-Kowal",
year = "2006",
month = "3",
doi = "10.1016/j.apnum.2005.04.021",
language = "English (US)",
volume = "56",
pages = "433--443",
journal = "Applied Numerical Mathematics",
issn = "0168-9274",
publisher = "Elsevier",
number = "3-4 SPEC. ISS.",

}

TY - JOUR

T1 - Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations

AU - Jackiewicz, Zdzislaw

AU - Zubik-Kowal, B.

PY - 2006/3

Y1 - 2006/3

N2 - We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated in a parallel computing environment by highly stable implicit methods. The effectiveness of this approach is illustrated by numerical experiments on the Hutchinson's equation. The boundedness of waveform relaxation iterations is proved for the Hutchinson's equation. This result is used in the proof of the superlinear convergence of the iterations.

AB - We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated in a parallel computing environment by highly stable implicit methods. The effectiveness of this approach is illustrated by numerical experiments on the Hutchinson's equation. The boundedness of waveform relaxation iterations is proved for the Hutchinson's equation. This result is used in the proof of the superlinear convergence of the iterations.

KW - Nonlinear partial delay differential equations

KW - Pseudospectral methods

KW - Waveform relaxation iterations

UR - http://www.scopus.com/inward/record.url?scp=33644611325&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644611325&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2005.04.021

DO - 10.1016/j.apnum.2005.04.021

M3 - Article

VL - 56

SP - 433

EP - 443

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 3-4 SPEC. ISS.

ER -