### Abstract

The long-term dynamics of a discretized, nonlinear, integro-differential equation with convolution kernel are studied. For a constant time-step algorithm the existence and stability of fixed and periodic points are investigated. A systematic treatment is given, which quantifies the effect of varying the quadrature rule and integrating the kernel exactly or approximately. Special attention is paid to spurious behaviour that occurs below, or around, the 'natural' time-step that corresponds to the linear stability limit for the correct fixed point. It is shown that spurious solutions exist, and can be computed, within this linear stability range. In addition to fixed points and period two solutions, analysis is performed for a class of period three orbits that are observed to be relevant to the long-term dynamics. Finally, an adaptive algorithm, based on local error control, is studied and a simple model describing its long-term behaviour is developed.

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

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Applied Numerical Mathematics |

Volume | 32 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2000 |

### Keywords

- convolution kernel
- error control
- linear stability
- time-stepping
- applied mathematics
- computer science

## Fingerprint Dive into the research topics of 'The effect of quadrature on the dynamics of a discretised nonlinear integro-differential equation'. Together they form a unique fingerprint.

## Cite this

Aves, M. A., Davies, P. J., & Higham, D. J. (2000). The effect of quadrature on the dynamics of a discretised nonlinear integro-differential equation.

*Applied Numerical Mathematics*,*32*(1), 1-20. https://doi.org/10.1016/S0168-9274(99)00013-6