An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data

Bangti Jin, Raytcho Lazarov, Zhi Zhou

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249 Scopus citations

Abstract

The subdiffusion equation with a Caputo fractional derivative of order α ε (0,1) in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for discretizing the Caputo fractional derivative in time. The scheme was analysed earlier independently by Lin and Xu (2007, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys., 225, 1533-1552) and Sun and Wu (2006, A fully discrete scheme for a diffusion wave system. Appl. Numer. Math., 56, 193-209), and an O(τ2-α) convergence rate was established, under the assumption that the solution is twice continuously differentiable in time. However, in view of the smoothing property of the subdiffusion equation, this regularity condition is restrictive, since it does not hold even for the homogeneous problem with a smooth initial data. In this work, we revisit the error analysis of the scheme, and establish an O(τ) convergence rate for both smooth and nonsmooth initial data. The analysis is valid for more general sectorial operators. In particular, the L1 scheme is applied to one-dimensional space-time fractional diffusion equations, which involves also a Riemann-Liouville derivative of order β ε (3/2,2) in space, and error estimates are provided for the fully discrete scheme. Numerical experiments are provided to verify the sharpness of the error estimates, and robustness of the scheme with respect to data regularity.
Original languageEnglish (US)
Pages (from-to)197-221
Number of pages25
JournalIMA Journal of Numerical Analysis
Volume36
Issue number1
DOIs
StatePublished - Jan 24 2015
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2022-05-26
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The research of BJ has been partially supported by NSF Grant DMS-1319052 and National Science Foundation of China No. 11471141, and that of RL was supported in parts by NSF Grant DMS-1016525 and also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). ZZ was partially supported by NSF Grant DMS-1016525.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • General Mathematics

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