Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 1173-1192.
Published online: 2022-10
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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} }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.