Volume 12, Issue 3
Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$, $\overline{{\rm L1}}$ and Alikhanov Schemes

East Asian J. Appl. Math., 12 (2022), pp. 503-520.

Published online: 2022-04

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• 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.

• Keywords

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

65L05, 65L70

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@Article{EAJAM-12-503, author = {}, title = {Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$, $\overline{{\rm L1}}$ and Alikhanov Schemes}, journal = {East Asian Journal on Applied Mathematics}, year = {2022}, volume = {12}, number = {3}, pages = {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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.290621.220921}, url = {http://global-sci.org/intro/article_detail/eajam/20402.html} }
TY - JOUR T1 - Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$, $\overline{{\rm L1}}$ and Alikhanov Schemes JO - East Asian Journal on Applied Mathematics VL - 3 SP - 503 EP - 520 PY - 2022 DA - 2022/04 SN - 12 DO - http://doi.org/10.4208/eajam.290621.220921 UR - https://global-sci.org/intro/article_detail/eajam/20402.html KW - ${\rm L1}$ scheme, $\overline{{\rm L1}}$ scheme, Alikhanov scheme, optimal convergence rate. AB -

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.

Yongtao Zhou & Martin Stynes. (2022). Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$, $\overline{{\rm L1}}$ and Alikhanov Schemes. East Asian Journal on Applied Mathematics. 12 (3). 503-520. doi:10.4208/eajam.290621.220921
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