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A New L1-TFPM Scheme for the Singularly Perturbed Subdiffusion Equations

A New $\mathcal{L}1$-TFPM Scheme for the Singularly Perturbed Subdiffusion Equations

Year:    2025

Author:    Wang Kong, Zhongyi Huang

CSIAM Transactions on Applied Mathematics, Vol. 6 (2025), Iss. 1 : pp. 1–30

Abstract

Since the memory effect is taken into account, the singularly perturbed subdiffusion equation can better describe the diffusion phenomenon with small diffusion coefficients. However, near the boundary configured with non-smooth boundary values, the solution of the singularly perturbed subdiffusion equation has a boundary layer of thickness O(ε), which brings great challenges to the construction of the efficient numerical schemes. By decomposing the Caputo fractional derivative, the singularly perturbed subdiffusion equation is formally transformed into a class of steady-state diffusive-reaction equation. By means of a kind of tailored finite point method (TFPM) scheme for solving steady-state diffusion-reaction equations and the L1 formula for discretizing the Caputo fractional derivative, we construct a new L1-TFPM scheme for solving singularly perturbed subdiffusion equations. Our proposed numerical scheme satisfies the discrete extremum principle and is unconditionally numerically stable. Besides, we prove that the new TFPM scheme can obtain reliable numerical solutions as hε and εh. However, there will be a large error loss due to the resonance effect as hε. Numerical experimental results can demonstrate the validity of the numerical scheme.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/csiam-am.SO-2023-0024

CSIAM Transactions on Applied Mathematics, Vol. 6 (2025), Iss. 1 : pp. 1–30

Published online:    2025-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    30

Keywords:    Singularly perturbed subdiffusion equations semi-discrete TFPM scheme L1-TFPM scheme discrete extremum principle.

Author Details

Wang Kong

Zhongyi Huang