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A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems

A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems
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Year:    2022

Author:    Martin Stynes

Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 4 : pp. 1173–1192

Abstract

A survey is given of convergence results that have been proved when the L1 scheme is used to approximate the Caputo time derivative $D^α_t$ (where $0 < α < 1)$ in initial-boundary value problems governed by $D^α_tu − ∆u = f$ and similar equations, while taking into account the weak singularity that is present in typical solutions of such problems. Various aspects of these analyses are outlined, such as global and local convergence bounds and the techniques used to derive them, fast implementation of the L1 scheme, semilinear problems, multi-term time derivatives, $α$-robustness, a posteriori error analysis, and two modified L1 schemes that achieve better accuracy. Over fifty references are provided in the bibliography, more than half of which are from the period 2019-2022.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.OA-2022-0009s

Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 4 : pp. 1173–1192

Published online:    2022-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    20

Keywords:    L1 scheme fractional derivative survey.

Author Details

Martin Stynes

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