Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$, $\overline{{\rm L1}}$ and Alikhanov Schemes
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:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18
Keywords: ${\rm L1}$ scheme $\overline{{\rm L1}}$ scheme Alikhanov scheme optimal convergence rate.
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