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Volume 15, Issue 4
A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems

Martin Stynes

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 1173-1192.

Published online: 2022-10

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  • 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.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-1173, author = {Stynes , Martin}, title = {A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0009s}, url = {http://global-sci.org/intro/article_detail/nmtma/21098.html} }
TY - JOUR T1 - A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems AU - Stynes , Martin JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1173 EP - 1192 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0009s UR - https://global-sci.org/intro/article_detail/nmtma/21098.html KW - L1 scheme, fractional derivative, survey. AB -

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.

Martin Stynes. (2022). A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 1173-1192. doi:10.4208/nmtma.OA-2022-0009s
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