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
Chien-Hong Cho,
Hisashi Okamoto
S. 391 - 415 doi:10.1553/etna_vol52s391 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol52s391
Abstract: We study finite difference schemes for axisymmetric blow-up solutions ofa nonlinear heat equation in higher spatial dimensions. The phenomenology of blow-up in higher-dimensional space is much more complex than that in one space dimension.To obtain a more complete picture for such phenomena from computational results, it is useful to know the technical details of the numerical schemes for higher spatial dimensions.Since first-order differentiation appears in the differential equation,we pay special attention to it. A sufficient condition for stabilityis derived. In addition to the convergence of the numerical blow-up time,certain blow-up behaviors, such as blow-up sets and blow-up in the $L^p$-norm,are taken into consideration. It is sometimes experienced that a certain property of solutions of a partial differential equation maybe lost by a faithfully constructed convergent numerical scheme. The phenomenon of one-point blow-up is a typical example in thenumerical analysis of blow-up problems. We prove that our scheme can preserve such a property.It is also remarkable that the $L^p$-norm $(1\leq p<\infty)$ of the solution of the nonlinear heat equation mayblow up simultaneously with the $L^\infty$-norm or remains bounded in $[0,T)$, where $T$ denotes the blow-uptime of the $L^\infty$-norm. We propose a systematic way to compute numerical evidence of the $L^p$-norm blow-up.The computational results are also analyzed. Moreover, we prove an abstract theorem which shows the relationshipbetween the numerical $L^p$-norm blow-up and the exact $L^p$-norm blow-up. Numerical examplesfor higher-dimensional blow-upsolutions are presented and discussed. Keywords: blow-up, finite difference method, nonlinear heat equation, $L^p$-norm blow-up Published Online: 2020/08/24 09:13:28 Object Identifier: 0xc1aa5572 0x003bc8be 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 |