Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$, $\overline{{\rm L1}}$ and Alikhanov Schemes

doi:10.4208/eajam.290621.220921

Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$ , $\overline{{\rm L1}}$ and Alikhanov Schemes

$Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$, $\overline{{\rm L1}}$ and Alikhanov Schemes$

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Year: 2022

East Asian Journal on Applied Mathematics, Vol. 12 (2022), Iss. 3 : pp. 503–520

Abstract

Consider the discretisation of the initial-value problem $D^αu(t) = f (t)$ for $0 < t \leq T$ with $u(0) = u_0$ , where $D^αu(t)$ is a Caputo derivative of order $α \in (0, 1)$ , using the ${\rm L1}$ scheme on a graded mesh with $N$ points. It is well known that one can prove the maximum nodal error in the computed solution is at most $\mathscr{O} (N^{− min\{rα,2−α\}})$ , where $r\geq 1$ is the mesh grading parameter ( $r = 1$ generates a uniform mesh). Numerical experiments indicate that this error bound is sharp, but no proof of its sharpness has been given. In the present paper the sharpness of this bound is proved, and the sharpness of the analogous nodal error bounds for the $\overline{{\rm L1}}$ and Alikhanov schemes will also be proved, using modifications of the ${\rm L1}$ analysis.

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Journal Article Details

Publisher Name: Global Science Press

Language: English

DOI: https://doi.org/10.4208/eajam.290621.220921

East Asian Journal on Applied Mathematics, Vol. 12 (2022), Iss. 3 : pp. 503–520

Published online: 2022-01

AMS Subject Headings:

Pages: 18

Keywords: ${\rm L1}$ scheme $\overline{{\rm L1}}$ scheme Alikhanov scheme optimal convergence rate.

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