The performance of preconditioned waveform relaxation techniques for pseudospectral methods

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8 Citations (Scopus)

Abstract

Waveform relaxation techniques for the pseudospectral solution of the heat conduction problem are discussed. The pseudospectral operator occurring in the equation is preconditioned by transformations in both space and time. In the spatial domain, domain stretching is used to more equally distribute the grid points across the domain, and hence improve the conditioning of the differential operator. Preconditioning in time is achieved either by an exponential or a polynomial transformation. Block Jacobi solutions of the systems are obtained and compared. The preconditioning in space by domain stretching determines the effectiveness of waveform relaxation in this case. Preconditioning in time is also effective in reducing the number of iterations required for convergence. The polynomial transformation is preferred, because it removes the requirement of a matrix exponential calculation in the time-stepping schemes and, at the same time, is no less effective than the direct exponential preconditioning.

Original languageEnglish (US)
Pages (from-to)245-263
Number of pages19
JournalNumerical Methods for Partial Differential Equations
Volume12
Issue number2
StatePublished - 1996

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Waveform Relaxation
Pseudospectral Method
Preconditioning
Stretching
Polynomials
Polynomial Transformation
Heat conduction
Matrix Exponential
Time Stepping
Heat Conduction
Conditioning
Jacobi
Differential operator
Grid
Iteration
Requirements
Operator

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Computational Mathematics

Cite this

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abstract = "Waveform relaxation techniques for the pseudospectral solution of the heat conduction problem are discussed. The pseudospectral operator occurring in the equation is preconditioned by transformations in both space and time. In the spatial domain, domain stretching is used to more equally distribute the grid points across the domain, and hence improve the conditioning of the differential operator. Preconditioning in time is achieved either by an exponential or a polynomial transformation. Block Jacobi solutions of the systems are obtained and compared. The preconditioning in space by domain stretching determines the effectiveness of waveform relaxation in this case. Preconditioning in time is also effective in reducing the number of iterations required for convergence. The polynomial transformation is preferred, because it removes the requirement of a matrix exponential calculation in the time-stepping schemes and, at the same time, is no less effective than the direct exponential preconditioning.",
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AU - Burrage, K.

AU - Jackiewicz, Zdzislaw

AU - Renaut, Rosemary

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AB - Waveform relaxation techniques for the pseudospectral solution of the heat conduction problem are discussed. The pseudospectral operator occurring in the equation is preconditioned by transformations in both space and time. In the spatial domain, domain stretching is used to more equally distribute the grid points across the domain, and hence improve the conditioning of the differential operator. Preconditioning in time is achieved either by an exponential or a polynomial transformation. Block Jacobi solutions of the systems are obtained and compared. The preconditioning in space by domain stretching determines the effectiveness of waveform relaxation in this case. Preconditioning in time is also effective in reducing the number of iterations required for convergence. The polynomial transformation is preferred, because it removes the requirement of a matrix exponential calculation in the time-stepping schemes and, at the same time, is no less effective than the direct exponential preconditioning.

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