Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation
Year: 2021
Author: Gundolf Haase, Reza Mokhtari, Mohadese Ramezani, Gundolf Haase
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 4 : pp. 945–971
Abstract
Some well-known L-type formulae, i.e., L1, L1-2, and L1-2-3 formulae, are usually employed to approximate the Caputo fractional derivative of order α ∈ (0, 1). In this paper, we aim to elaborate on the stability and convergence analyses of some finite difference methods (FDMs) for solving the subdiffusion equation, i.e., a diffusion equation which exploits the Caputo time-fractional derivative of order $α$. In fact, the FDMs considered here are based on the usual central difference scheme for the spatial derivative, and the Caputo derivative is approximated by using methods such as the L1, L1-2, and L1-2-3 formulae. Thanks to a specific type of the discrete version of the Gronwall inequality, we show that the FDMs are unconditionally stable in the maximum norm and also discrete $H^1$ norm. Then, we prove that the finite difference method which uses the L1, L1-2, and L1-2-3 formulae has the global order of convergence $2−α$, $3−α$, and 3, respectively. Finally, some numerical tests confirm the theoretical results. A brief conclusion finishes the paper.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2021-0020
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 4 : pp. 945–971
Published online: 2021-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
Keywords: Stability analysis order of convergence Caputo derivative L1 formula L1-2 formula L1-2-3 formula subdiffusion equation Gronwall inequality.
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