A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations
type: Item
instance of: scholarly article
title: A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations
zbMATH Open document ID: 1379.65056
author: Benjamin Kehlet
author: Anders Logg
published in: Numerical Algorithms
publication date: 2017-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1504.06765
review text: The global error for the numerical solution of ordinary differential equations is comprised of errors which originate from a variety of sources. In this work the authors extend these standard estimates by adding a new term accounting for the use of finite numeric precision in the computation of the numerical solution. This error is normally neglected due to its magnitude being much smaller than the contributions provided by the data or discretization error. Since general initial value problems for systems of ordinary differential equations can be very sensitive to perturbations when the time interval \([0, T]\) is very long, or when a solution is sought with very high accuracy, it is important that the effect of numerical round-off errors as a result of finite numeric precision is considered. This will inevitably be the dominating error source which will limit the computability of a given problem if not taken into account. Examples are investigated to support the analysis provided: the Lorenz system and Van der Pol oscillator. Both examples provide support for the work conducted by displaying competing convergence rates for discretisation errors, decreasing rapidly for smaller time steps, and computational errors (round-off error), increasing for smaller time steps. The authors note that ``the effect of round-off errors is mostly pronounced for large values of the stability factor, which includes both chaotic dynamical systems as well as long-time integration of systems which exhibit only a moderate growth of the stability factor.''
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65L20
zbMATH DE Number: 6788833
zbMATH Keywords: computability
zbMATH Keywords: high precision
zbMATH Keywords: high order
zbMATH Keywords: high accuracy
zbMATH Keywords: probabilistic error propagation
zbMATH Keywords: long-time integration
zbMATH Keywords: finite element
zbMATH Keywords: time-stepping
zbMATH Keywords: a posteriori error bound
zbMATH Keywords: Lorenz system
zbMATH Keywords: Van der Pol oscillator
zbMATH Keywords: convergence
zbMATH Keywords: round-off error
zbMATH Keywords: stability
zbMATH Keywords: chaotic dynamical systems
Wikidata QID: Q59610820
reviewed by: Charis Harley
describes a project that uses: deTestSet
describes a project that uses: Taylor
describes a project that uses: mctoolbox
describes a project that uses: gmp
MaRDI profile type: Publication
OpenAlex ID: W1628431942
arXiv ID: 1504.06765
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DBLP publication ID: journals/na/KehletL17
DOI: 10.1007/S11075-016-0250-4
description: scientific article