An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions

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dc.contributor.authorChen, Hu
dc.contributor.authorHolland, Finbarr
dc.contributor.authorStynes, Martin
dc.contributor.funderChina Postdoctoral Science Foundation
dc.contributor.funderNational Natural Science Foundation of China
dc.date.accessioned2019-02-12T10:25:16Z
dc.date.available2019-02-12T10:25:16Z
dc.date.issued2019-01-15
dc.date.updated2019-02-12T10:06:07Z
dc.description.abstractA convergence analysis is given for the Grünwald–Letnikov discretisation of a Riemann–Liouville fractional initial-value problem on a uniform mesh tm=mτ with m=0,1,…,M. For given smooth data, the unknown solution of the problem will usually have a weak singularity at the initial time t=0. Our analysis is the first to prove a convergence result for this method while assuming such non-smooth behaviour in the unknown solution. In part our study imitates previous analyses of the L1 discretisation of such problems, but the introduction of some additional ideas enables exact formulas for the stability multipliers in the Grünwald–Letnikov analysis to be obtained (the earlier L1 analyses yielded only estimates of their stability multipliers). Armed with this information, it is shown that the solution computed by the Grünwald–Letnikov scheme is O(τtmα−1) at each mesh point tm; hence the scheme is globally only O(τα) accurate, but it is O(τ) accurate for mesh points tm that are bounded away from t=0. Numerical results for a test example show that these theoretical results are sharp.en
dc.description.sponsorshipChina Postdoctoral Science Foundation (Grant 2018M631316); National Natural Science Foundation of China (Grants 11801026; 91430216; NSAF-U1530401)
dc.description.statusPeer revieweden
dc.description.versionAccepted Versionen
dc.format.mimetypeapplication/pdfen
dc.identifier.citationChen, H., Holland, F. and Stynes, M. (2019) 'An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions', Applied Numerical Mathematics, 139, pp. 52-61. doi:10.1016/j.apnum.2019.01.004en
dc.identifier.doi10.1016/j.apnum.2019.01.004
dc.identifier.endpage61en
dc.identifier.issn0168-9274
dc.identifier.issn1873-5460
dc.identifier.journaltitleApplied Numerical Mathematicsen
dc.identifier.startpage52en
dc.identifier.urihttps://hdl.handle.net/10468/7479
dc.identifier.volume139en
dc.language.isoenen
dc.publisherElsevier B.V.en
dc.rights© 2019, IMACS. Published by Elsevier B.V. All rights reserved. This manuscript version is made available under the CC BY-NC-ND 4.0 license.en
dc.rights.urihttps://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectWeak singularityen
dc.subjectConvergence analysisen
dc.subjectRiemann–Liouville derivativeen
dc.subjectGrünwald–Letnikov schemeen
dc.titleAn analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutionsen
dc.typeArticle (peer-reviewed)en
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