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 H1 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.
Author Details
Gundolf Haase Email
Reza Mokhtari Email
Mohadese Ramezani Email
Gundolf Haase Email
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