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19-08-2021

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

Authors: Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang

Published in: BIT Numerical Mathematics | Issue 2/2022

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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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Title: Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes
Authors: Baoli Yin
Yang Liu
Hong Li
Zhimin Zhang
Publication date: 19-08-2021
Publisher: Springer Netherlands
Published in: BIT Numerical Mathematics / Issue 2/2022
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-021-00890-z

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