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Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions

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Abstract

Thanks to the singularity of the solution of linear subdiffusion problems, most time-step** methods on uniform meshes can result in \(O(\tau )\) accuracy where \(\tau \) denotes the time step. The present work aims to discover the reason why some type of Crank-Nicolson schemes (the averaging Crank-Nicolson (ACN) scheme) for the subdiffusion can only yield \(O(\tau ^\alpha )\,\, (\alpha <1)\) accuracy, which is much lower than the desired. The existing well-developed error analysis for the subdiffusion, which has been successfully applied to many time-step** methods such as the fractional BDF-\(p\,\,(1\leqslant p\leqslant 6)\), requires singular points to be out of the path of contour integrals involved. The ACN scheme in this work is quite natural but fails to meet this requirement. By resorting to the residue theorem, some novel sharp error analysis is developed in this study, upon which correction methods are further designed to obtain the optimal \(O(\tau ^2)\) accuracy. All results are verified by numerical tests.

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Data Availability

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

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Acknowledgements

This work is supported by the Autonomous Region Level High-Level Talent Introduction Research Support Program in 2022 (No. 12000-15042224 to B.Y.), the National Natural Science Foundation of China (No. 12201322 to B.Y., 12061053 to Y.L., and 12161063 to H.L.), and the Natural Science Foundation of Inner Mongolia, China (No. 2020MS01003 to Y.L. and 2021MS01018 to H.L.).

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  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, Inner Mongolia, China

    Baoli Yin, Yang Liu & Hong Li

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  1. Baoli Yin
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Correspondence to Hong Li.

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Yin, B., Liu, Y. & Li, H. Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-024-00401-1

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  • DOI: https://doi.org/10.1007/s42967-024-00401-1

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