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Foundations of Computational Mathematics

Erschienen in:

01.04.2015

Higher-Order Averaging, Formal Series and Numerical Integration III: Error Bounds

verfasst von: P. Chartier, A. Murua, J. M. Sanz-Serna

Erschienen in: Foundations of Computational Mathematics | Ausgabe 2/2015

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Abstract

In earlier papers, it has been shown how formal series like those used nowadays to investigate the properties of numerical integrators may be used to construct high-order averaged systems or formal first integrals of Hamiltonian problems. With the new approach the averaged system (or the formal first integral) may be written down immediately in terms of (i) suitable basis functions and (ii) scalar coefficients that are computed via simple recursions. Here we show how the coefficients/basis functions approach may be used advantageously to derive exponentially small error bounds for averaged systems and approximate first integrals.

Vorheriger Artikel Nonlinear Approximation Rates and Besov Regularity for Elliptic PDEs on Polyhedral Domains

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Note that since the function f only enters the analysis through the bounds in Assumption A below, it is possible to allow f=f(y,θ,ϵ) provided that Assumption A holds with ϵ-independent constants.

Throughout the paper it is understood that for n=1 the expression (n−1)^a(n−1), a>0 takes the value 1.

For a proof note that, according to the theory of continued fractions, |qω−p|(p+q), p, q positive integers, is ‘small’ when p/q is a convergent of the continued fraction for ω, i.e. when q and p are consecutive Fibonacci numbers \(\mathcal {F}_{j}\), \(\mathcal {F}_{j+1}\). By using the closed-form expression for the \(\mathcal {F}_{j}\), it is a tedious exercise to show that \(|\mathcal {F}_{j}\omega-\mathcal {F}_{j+1}| (\mathcal {F}_{j}+\mathcal {F}_{j+1})>1\). Note that the value ν=1 is exceptionally low, because, as pointed out before, most ω satisfy (28) only for ν>d−1.

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Titel: Higher-Order Averaging, Formal Series and Numerical Integration III: Error Bounds
verfasst von: P. Chartier
A. Murua
J. M. Sanz-Serna
Publikationsdatum: 01.04.2015
Verlag: Springer US
Erschienen in: Foundations of Computational Mathematics / Ausgabe 2/2015
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-013-9175-7

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