### Abstract

Language | English |
---|---|

Pages | 227-236 |

Number of pages | 9 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 45 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 8 Apr 1993 |

### Fingerprint

### Keywords

- global error
- interpolation
- tolerance proportionality
- delay ordinary differential equations
- differential equations

### Cite this

*Journal of Computational and Applied Mathematics*,

*45*(1-2), 227-236. https://doi.org/10.1016/0377-0427(93)90277-I

}

*Journal of Computational and Applied Mathematics*, vol. 45, no. 1-2, pp. 227-236. https://doi.org/10.1016/0377-0427(93)90277-I

**The tolerance proportionality of adaptive ODE solvers.** / Higham, D.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The tolerance proportionality of adaptive ODE solvers

AU - Higham, D.J.

PY - 1993/4/8

Y1 - 1993/4/8

N2 - Modern software for solving ordinary differential equation (ODE) initial-value problems requires the user to specify the ODE and choose a value for the error tolerance. The software can be thought of as a black box with a dial - turning the dial changes the accuracy and expense of the integration process. It is therefore of interest to know how the global error varies with the error tolerance. In this work, we look at explicit Runge-Kutta methods and show that with any standard error control method, and ignoring higher-order terms, the global error in the numerical solution behaves like a known rational power of the error tolerance. This generalises earlier work of Stetter, who found sufficient conditions for the global error to be linear in the tolerance. We also display the order of the next-highest term. We then analyse continuous Runge-Kutta schemes, and show what order of interpolation is necessary and sufficient for the continuous approximation to inherit the tolerance proportionality of the discrete formula. Finally we extend the results to the case of ODE systems with constant delays, thereby generalising some previous results of the author.

AB - Modern software for solving ordinary differential equation (ODE) initial-value problems requires the user to specify the ODE and choose a value for the error tolerance. The software can be thought of as a black box with a dial - turning the dial changes the accuracy and expense of the integration process. It is therefore of interest to know how the global error varies with the error tolerance. In this work, we look at explicit Runge-Kutta methods and show that with any standard error control method, and ignoring higher-order terms, the global error in the numerical solution behaves like a known rational power of the error tolerance. This generalises earlier work of Stetter, who found sufficient conditions for the global error to be linear in the tolerance. We also display the order of the next-highest term. We then analyse continuous Runge-Kutta schemes, and show what order of interpolation is necessary and sufficient for the continuous approximation to inherit the tolerance proportionality of the discrete formula. Finally we extend the results to the case of ODE systems with constant delays, thereby generalising some previous results of the author.

KW - global error

KW - interpolation

KW - tolerance proportionality

KW - delay ordinary differential equations

KW - differential equations

U2 - 10.1016/0377-0427(93)90277-I

DO - 10.1016/0377-0427(93)90277-I

M3 - Article

VL - 45

SP - 227

EP - 236

JO - Journal of Computational and Applied Mathematics

T2 - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-2

ER -