TY - JOUR
T1 - Optimal Convergence Rates in Time-Fractional Discretisations: The ${\rm L1}$, $\overline{{\rm L1}}$ and Alikhanov Schemes
AU - Yongtao Zhou & Martin Stynes
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 - <p style="text-align: justify;">Consider the discretisation of the initial-value problem $D^αu(t) = f (t)$ for $0 &lt; t \leq T$ 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.</p>