### Abstract

Original language | English |
---|---|

Pages (from-to) | 151-169 |

Number of pages | 18 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 58 |

Issue number | 2 |

DOIs | |

Publication status | Published - 30 Mar 1995 |

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### Keywords

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

### Cite this

*Journal of Computational and Applied Mathematics*,

*58*(2), 151-169. https://doi.org/10.1016/0377-0427(93)E0268-Q

}

*Journal of Computational and Applied Mathematics*, vol. 58, no. 2, pp. 151-169. https://doi.org/10.1016/0377-0427(93)E0268-Q

**Equilibrium states of adaptive algorithms for delay differential equations.** / Higham, D.J.; Famelis, I.T.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Equilibrium states of adaptive algorithms for delay differential equations

AU - Higham, D.J.

AU - Famelis, I.T.

PY - 1995/3/30

Y1 - 1995/3/30

N2 - 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.

AB - 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.

KW - Runge-Kutta method

KW - Error control

KW - Fixed point

KW - Delay

KW - numerical mathematics

UR - http://dx.doi.org/10.1016/0377-0427(93)E0268-Q

U2 - 10.1016/0377-0427(93)E0268-Q

DO - 10.1016/0377-0427(93)E0268-Q

M3 - Article

VL - 58

SP - 151

EP - 169

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 2

ER -