A pseudospectral tau approximation for time delay systems and its comparison with other weighted-residual-type methods

Dávid Lehotzky, T. Insperger

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

This paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations, which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams for RFDEs. Thorough numerical case studies show that the rightmost characteristic roots of the ordinary differential equation approximation converge to the rightmost characteristic roots of the original RFDE. Application of the method to time-periodic RFDEs is also demonstrated, and the convergence of the stability boundaries is verified numerically. The method is compared with recently developed highly efficient numerical methods: the pseudospectral collocation (also called Chebyshev spectral continuous-time approximation), the spectral Legendre tau method, and the spectral element method. The comparison is based on the stability analysis of three linear autonomous RFDEs. The efficiency of the methods is measured by the convergence rate of stability boundaries in the space of system parameters, by the convergence rate of the rightmost characteristic exponent and by the computation time of the stability charts.

Original languageEnglish
JournalInternational Journal for Numerical Methods in Engineering
DOIs
Publication statusAccepted/In press - 2016

Fingerprint

Retarded Functional Differential Equations
Time-delay Systems
Time delay
Differential equations
Approximation
Characteristic Roots
Ordinary differential equations
Rate of Convergence
Tau Method
Spectral Element Method
Finite-dimensional Approximation
Characteristic Exponents
Legendre
Chebyshev
Collocation
System of Ordinary Differential Equations
Chart
Continuous Time
Stability Analysis
Numerical methods

Keywords

  • Linear stability
  • Numerical method
  • Tau method
  • Time delay
  • Weighted residual method

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics
  • Numerical Analysis

Cite this

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abstract = "This paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations, which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams for RFDEs. Thorough numerical case studies show that the rightmost characteristic roots of the ordinary differential equation approximation converge to the rightmost characteristic roots of the original RFDE. Application of the method to time-periodic RFDEs is also demonstrated, and the convergence of the stability boundaries is verified numerically. The method is compared with recently developed highly efficient numerical methods: the pseudospectral collocation (also called Chebyshev spectral continuous-time approximation), the spectral Legendre tau method, and the spectral element method. The comparison is based on the stability analysis of three linear autonomous RFDEs. The efficiency of the methods is measured by the convergence rate of stability boundaries in the space of system parameters, by the convergence rate of the rightmost characteristic exponent and by the computation time of the stability charts.",
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AU - Insperger, T.

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N2 - This paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations, which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams for RFDEs. Thorough numerical case studies show that the rightmost characteristic roots of the ordinary differential equation approximation converge to the rightmost characteristic roots of the original RFDE. Application of the method to time-periodic RFDEs is also demonstrated, and the convergence of the stability boundaries is verified numerically. The method is compared with recently developed highly efficient numerical methods: the pseudospectral collocation (also called Chebyshev spectral continuous-time approximation), the spectral Legendre tau method, and the spectral element method. The comparison is based on the stability analysis of three linear autonomous RFDEs. The efficiency of the methods is measured by the convergence rate of stability boundaries in the space of system parameters, by the convergence rate of the rightmost characteristic exponent and by the computation time of the stability charts.

AB - This paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations, which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams for RFDEs. Thorough numerical case studies show that the rightmost characteristic roots of the ordinary differential equation approximation converge to the rightmost characteristic roots of the original RFDE. Application of the method to time-periodic RFDEs is also demonstrated, and the convergence of the stability boundaries is verified numerically. The method is compared with recently developed highly efficient numerical methods: the pseudospectral collocation (also called Chebyshev spectral continuous-time approximation), the spectral Legendre tau method, and the spectral element method. The comparison is based on the stability analysis of three linear autonomous RFDEs. The efficiency of the methods is measured by the convergence rate of stability boundaries in the space of system parameters, by the convergence rate of the rightmost characteristic exponent and by the computation time of the stability charts.

KW - Linear stability

KW - Numerical method

KW - Tau method

KW - Time delay

KW - Weighted residual method

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