Equilibrium states of adaptive algorithms for delay differential equations

D.J. Higham, I.T. Famelis

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

This work examines the performance of explicit, adaptive, Runge-Kutta based algorithms for solving delay differential equations. The results of Hall (1985) for ordinary differential equation (ODE) solvers are extended by adding a constant-delay term to the test equation. It is shown that by regarding an algorithm as a discrete nonlinear map, fixed points or equilibrium states can be identified and their stability can be determined numerically. Specific results are derived for a low order Runge-Kutta pair coupled with either a linear or cubic interpolant. The qualitative performance is shown to depend upon the interpolation process, in addition to the ODE formula and the error control mechanism. Furthermore, and in contrast to the case for standard ODEs, it is found that the parameters in the test equation also influence the behaviour. This phenomenon has important implications for the design of robust algorithms. The choice of error tolerance, however, is shown not to affect the stability of the equilibrium states. Numerical tests are used to illustrate the analysis. Finally, a general result is given which guarantees the existence of equilibrium states for a large class of algorithms.
Original languageEnglish
Pages (from-to)151-169
Number of pages18
JournalJournal of Computational and Applied Mathematics
Volume58
Issue number2
DOIs
Publication statusPublished - 30 Mar 1995

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Adaptive algorithms
Delay Differential Equations
Adaptive Algorithm
Equilibrium State
Differential equations
Runge-Kutta
Ordinary differential equation
Ordinary differential equations
Nonlinear Map
Robust Algorithm
Error Control
Interpolants
Tolerance
Interpolate
Fixed point
Interpolation
Term

Keywords

  • Runge-Kutta method
  • Error control
  • Fixed point
  • Delay
  • numerical mathematics

Cite this

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abstract = "This work examines the performance of explicit, adaptive, Runge-Kutta based algorithms for solving delay differential equations. The results of Hall (1985) for ordinary differential equation (ODE) solvers are extended by adding a constant-delay term to the test equation. It is shown that by regarding an algorithm as a discrete nonlinear map, fixed points or equilibrium states can be identified and their stability can be determined numerically. Specific results are derived for a low order Runge-Kutta pair coupled with either a linear or cubic interpolant. The qualitative performance is shown to depend upon the interpolation process, in addition to the ODE formula and the error control mechanism. Furthermore, and in contrast to the case for standard ODEs, it is found that the parameters in the test equation also influence the behaviour. This phenomenon has important implications for the design of robust algorithms. The choice of error tolerance, however, is shown not to affect the stability of the equilibrium states. Numerical tests are used to illustrate the analysis. Finally, a general result is given which guarantees the existence of equilibrium states for a large class of algorithms.",
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Equilibrium states of adaptive algorithms for delay differential equations. / Higham, D.J.; Famelis, I.T.

In: Journal of Computational and Applied Mathematics, Vol. 58, No. 2, 30.03.1995, p. 151-169.

Research output: Contribution to journalArticle

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AU - Higham, D.J.

AU - Famelis, I.T.

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