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

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 $\mathcal{O}\left(\epsilon \right)$, 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 steadystate 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\ll \epsilon $ and $\epsilon \ll h$. However, there will be a large error loss due to the resonance effect as $h\sim \epsilon $. Numerical experimental results can demonstrate the validity of the numerical scheme.

Keywords

Singularly perturbed subdiffusion equations / semi-discrete TFPM scheme / $\mathcal{L}1-$ TFPM scheme / discrete extremum principle

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