ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis ISBN 978-3-7001-8258-0 Online Edition Research Article
Miljenko Marušić
S. 329 - 347 doi:10.1553/etna_vol48s329 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol48s329
Abstract: We introduce a family of exponentially fitted difference schemes of arbitrary orderas numerical approximations to the solution of a singularly perturbed two-point boundary valueproblem: $\\varepsilon y” + b y\' + c y = f$.The difference schemes are derived from interpolation formulae for exponential sums.The so-defined $k$-point differentiation formulae are exact for functions that are alinear combination of $1,x,\\ldots,x^{k-2},\\exp{(-\\rho x)}$.The parameter $\\rho$ is chosen from the asymptotic behavior of the solution in the boundary layer.This approach allows a construction of the method with arbitrary order of consistency.Using an estimate for the interpolation error, we prove consistency of all the schemes fromthe family.The truncation error is bounded by $C h^{k-2}$, where $C$ is a constant independent of $\\varepsilon$and $h$.Therefore, the order of consistency for the $k$-point scheme is $k-2$ ($k \\geq 3$) in case ofa small perturbation parameter $\\varepsilon$.There is no general proof of stability for the proposed schemes.Each scheme has to be considered separately.In the paper, stability, and therefore convergence, is proved for three-point schemesin the case when $c<0$ and $b \\neq 0$. Keywords: difference scheme, singular perturbation, ODE, interpolation, exponential sum Published Online: 2018/09/10 14:01:04 Object Identifier: 0xc1aa5576 0x0039e77e Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613. …
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |