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Journal of Scientific Computing

, Volume 79, Issue 1, pp 624–647 | Cite as

Error Analysis of a Second-Order Method on Fitted Meshes for a Time-Fractional Diffusion Problem

  • Hu Chen
  • Martin StynesEmail author
Article

Abstract

Alikhanov’s high-order scheme for Caputo fractional derivatives of order \(\alpha \in (0,1)\) is generalised to nonuniform meshes and analysed for initial-value problems (IVPs) and initial-boundary value problems (IBVPs) whose solutions display a typical weak singularity at the initial time. It is shown that, when the mesh is chosen suitably, the scheme attains order \(3-\alpha \) convergence for the 1-dimensional IVP and second-order convergence for the IBVP, for which a spectral method is analysed when the spatial domain is the unit square and the extension of this analysis to other spatial domains and other spatial dimensions and discretisations is outlined. Numerical results demonstrate the sharpness of the theoretical convergence estimates.

Keywords

Fractional differential equation Initial-boundary value problem Weak singularity Alikhanov scheme Graded mesh 

Mathematics Subject Classification

65M06 65M12 65M60 65M70 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied and Computational Mathematics DivisionBeijing Computational Science Research CenterBeijingChina

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