TY - JOUR
T1 - A New $\mathcal{L}1$-TFPM Scheme for the Singularly Perturbed Subdiffusion Equations
AU - Kong , Wang
AU - Huang , Zhongyi
JO - CSIAM Transactions on Applied Mathematics
VL - 1
SP - 1
EP - 30
PY - 2025
DA - 2025/02
SN - 6
DO - http://doi.org/10.4208/csiam-am.SO-2023-0024
UR - https://global-sci.org/intro/article_detail/csiam-am/23794.html
KW - Singularly perturbed subdiffusion equations, semi-discrete TFPM scheme, $\mathcal{L}1$-TFPM scheme, discrete extremum principle.
AB - <p style="text-align: justify;">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 $\mathcal{O}(ε),$ 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 $\mathcal{L}1$ formula for discretizing the Caputo fractional derivative, we construct a new $\mathcal{L}1$-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 $h ≪ ε$ and $ε ≪ h.$ However, there will be a large error loss due
to the resonance effect as $h ∼ ε.$ Numerical experimental results can demonstrate the
validity of the numerical scheme.</p>