Adaptive time-stepping and computational stability.

*(English)*Zbl 1077.65086Summary: We investigate the effects of adaptive time-stepping and other algorithmic strategies on the computational stability of ordinary differential equation codes. We show that carefully designed adaptive algorithms have a most significant impact on computational stability and reliability. A series of computational experiments with the standard implementation of DASSL and a modified version, including stepsize control based on digital filters, is used to demonstrate that relatively small algorithmic changes are able to extract a vastly better computational stability at no extra expense. The inherent performance and stability of DASSL are therefore much greater than the standard implementation seems to suggest.

##### MSC:

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65L80 | Numerical methods for differential-algebraic equations |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

##### Keywords:

Computational stability; Stepsize control; Adaptive time-stepping; PI control; Digital filters; DASSL; Mathematical software; Algorithm analysis; numerical examples; differential-algebraic equations; adaptive algorithms; performance
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\textit{G. Söderlind} and \textit{L. Wang}, J. Comput. Appl. Math. 185, No. 2, 225--243 (2006; Zbl 1077.65086)

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