The effects of weak hyperbolicity on the diffusion of heat

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article King A. C., Needham D. J. and Scott N. H. 1998The effects of weak hyperbolicity on the diffusion of heatProc. R. Soc. Lond. A.4541659–1679http://doi.org/10.1098/rspa.1998.0225SectionRestricted accessThe effects of weak hyperbolicity on the diffusion of heat A. C. King A. C. King School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK Google Scholar Find this author on PubMed Search for more papers by this author , D. J. Needham D. J. Needham Department of Mathematics, University of Reading, Reading RG6 6AX, UK Google Scholar Find this author on PubMed Search for more papers by this author and N. H. Scott N. H. Scott School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK Google Scholar Find this author on PubMed Search for more papers by this author A. C. King A. C. King School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK Google Scholar Find this author on PubMed Search for more papers by this author , D. J. Needham D. J. Needham Department of Mathematics, University of Reading, Reading RG6 6AX, UK Google Scholar Find this author on PubMed Search for more papers by this author and N. H. Scott N. H. Scott School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK Google Scholar Find this author on PubMed Search for more papers by this author Published:08 June 1998https://doi.org/10.1098/rspa.1998.0225AbstractThe behaviour of the weakly hyperbolic partial differential equation εUtt+Ut=Uxx, in which ϵ < 0 is a small parameter is compared with that of the parabolic diffusion equation for which ϵ = 0. It is well known that for ϵ = 0 part of an initially localized disturbance reaches infinity instantaneously whereas for 0 < ϵ ≪ 1 a small part of the disturbance is propagated out into an undisturbed region by means of waves with large propagation speeds ±√1/ϵ which have amplitudes heavily damped by the factor e–t/2ϵ. It is shown here that the bulk of the disturbance diffuses out as though ϵ = 0 except that the infinite tails are clipped off by the wave fronts described above. It is also shown that for t ≫ O(ϵ) two further asymptotic regions are needed to connect the diffusive bulk with the decaying wave fronts. Existing numerical methods for hyperbolic and for parabolic equations do not work well on (A) and so a hybrid method is developed and its convergence and stability considered. The foregoing discussion of (A) is couched in terms of heat conduction but the equation may also be obtained in chemical kinetics if a new constitutive assumption linking the flux of a chemical species and its rate of change with the concentration gradient is posited, see equation 1.1. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Ozorio Cassol G and Dubljevic S (2021) Hyperbolicity of reaction‐transport processes, AIChE Journal, 10.1002/aic.17135, 67:4, Online publication date: 1-Apr-2021. Leach J and Bassom A (2019) Long-time solutions of scalar nonlinear hyperbolic reaction equations incorporating relaxation I. The reaction function is a bistable cubic polynomial, Journal of Differential Equations, 10.1016/j.jde.2018.07.077, 266:2-3, (1285-1312), Online publication date: 1-Jan-2019. 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Fedotov S (2000) Front dynamics for an anisotropic reaction-diffusion equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/33/40/302, 33:40, (7033-7042), Online publication date: 13-Oct-2000. This Issue08 June 1998Volume 454Issue 1974 Article InformationDOI:https://doi.org/10.1098/rspa.1998.0225Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/06/1998Published in print08/06/1998 License: Citations and impact Keywordsdiffusionhyperbolicity