### Abstract

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

Pages | 1-20 |

Number of pages | 20 |

Journal | Applied Numerical Mathematics |

Volume | 32 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2000 |

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### Keywords

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

### Cite this

*Applied Numerical Mathematics*,

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

}

*Applied Numerical Mathematics*, vol. 32, no. 1, pp. 1-20. https://doi.org/10.1016/S0168-9274(99)00013-6

**The effect of quadrature on the dynamics of a discretised nonlinear integro-differential equation.** / Aves, M.A.; Davies, P.J.; Higham, D.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The effect of quadrature on the dynamics of a discretised nonlinear integro-differential equation

AU - Aves, M.A.

AU - Davies, P.J.

AU - Higham, D.J.

PY - 2000/1

Y1 - 2000/1

N2 - 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.

AB - 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.

KW - convolution kernel

KW - error control

KW - linear stability

KW - time-stepping

KW - applied mathematics

KW - computer science

U2 - 10.1016/S0168-9274(99)00013-6

DO - 10.1016/S0168-9274(99)00013-6

M3 - Article

VL - 32

SP - 1

EP - 20

JO - Applied Numerical Mathematics

T2 - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 1

ER -