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Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data
Yan, Yubin Wang, Yanyong Yan, Yuyuan Pani, Amiya K.
Yan, Yubin
Wang, Yanyong
Yan, Yuyuan
Pani, Amiya K.
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2020-05-19
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Abstract
Two higher order time stepping methods for solving subdiffusion problems are studied in this paper.
The Caputo time fractional derivatives are approximated by using the weighted and shifted Gr\"unwald-Letnikov formulae
introduced in Tian et al. [Math. Comp. 84 (2015), pp. 2703-2727]. After correcting a few starting steps, the proposed time
stepping methods have the optimal convergence orders $O(k^2)$ and $ O(k^3)$, respectively for any fixed time $t$ for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of
our time stepping schemes to various other fractional evolution equations. Finally,
some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.
Citation
Wang, Y., Yan, Y., Yan, Y. et al. (2020). Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data. Journal of Scientific Computing, 83, 40.
Publisher
Springer
Journal
Journal of Scientific Computing
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DOI
10.1007/s10915-020-01223-y
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Article
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1573-7691