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Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes

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Abstract

This article devotes to developing robust and simple correction techniques with efficient algorithms for a class of second-order time stepping methods, namely the shifted fractional trapezoidal rule (SFTR), for subdiffusion problems to resolve the initial singularity and nonlocality. The stability analysis and sharp error estimates in terms of the smoothness of the initial data and source term are presented. As a byproduct in numerical tests, we find amazingly that the Crank–Nicolson scheme (\(\theta =\frac{1}{2}\)) without initial corrections can restore the optimal convergence rate for the subdiffusion problem with smooth initial data and source terms. To deal with the nonlocality, fast algorithms are considered to reduce the computational cost from \(O(N^2)\) to \(O(N \log N)\) and save the memory storage from O(N) to \(O(\log N)\), where N denotes the number of time levels. Numerical tests are performed to verify the sharpness of the theoretical results and confirm the efficiency and accuracy of initial corrections and the fast algorithms.

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Authors and Affiliations

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

    Baoli Yin, Yang Liu & Hong Li

  2. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

  3. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

Authors
  1. Baoli Yin
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  2. Yang Liu
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  3. Hong Li
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  4. Zhimin Zhang
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Correspondence to Yang Liu.

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Communicated by Mihaly Kovacs.

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The work of the second author was supported in part by Grants NSFC 12061053, 11661058 and the NSF of Inner Mongolia 2020MS01003. The work of the third author was supported in part by the Grant NSFC 11761053. The work of the fourth author was supported in part by Grants NSFC 11871092 and NSAF U1930402.

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Yin, B., Liu, Y., Li, H. et al. Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes. Bit Numer Math 62, 631–666 (2022). https://doi.org/10.1007/s10543-021-00890-z

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  • DOI: https://doi.org/10.1007/s10543-021-00890-z

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