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An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions
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Accepted Version
Date
2019-01-15
Authors
Chen, Hu
Holland, Finbarr
Stynes, Martin
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier B.V.
Published Version
Abstract
A 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.
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Keywords
Weak singularity , Convergence analysis , Riemann–Liouville derivative , Grünwald–Letnikov scheme
Citation
Chen, 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.2019.01.004