$\alpha$-robust error estimates of general non-uniform time-step numerical schemes for reaction-subdiffusion problems

TL;DR

This paper develops an $ ext{alpha}$-robust error analysis for convolution-type numerical schemes solving reaction-subdiffusion problems, ensuring error bounds remain stable as the fractional order approaches 1, even with nonuniform time steps.

Contribution

It provides the first explicit $ ext{alpha}$-robust error bounds for general nonuniform time-step schemes applied to subdiffusion equations, addressing a key limitation of prior analyses.

Findings

Error bounds do not blow up as $ ext{alpha} o 1^-$.

The stability and convergence are $ ext{alpha}$-robust for the L1 and Alikhanov schemes.

Explicit dependence on $ ext{alpha}$ and mesh sizes is established.

Abstract

Numerous error estimates have been carried out on various numerical schemes for subdiffusion equations. Unfortunately most error bounds suffer from a factor $1/(1-\alpha)$ or $\Gamma(1-\alpha)$, which blows up as the fractional order $\alpha\to 1^-$, a phenomenon not consistent with regularity of the continuous problem and numerical simulations in practice. Although efforts have been made to avoid the factor blow-up phenomenon, a robust analysis of error estimates still remains incomplete for numerical schemes with general nonuniform time steps. In this paper, we will consider the $\alpha$-robust error analysis of convolution-type schemes for subdiffusion equations with general nonuniform time-steps, and provide explicit factors in error bounds with dependence information on $\alpha$ and temporal mesh sizes. As illustration, we apply our abstract framework to two widely used schemes, i.e., the L1 scheme and Alikhanov's scheme. Our rigorous proofs reveal that the stability and convergence of a class of convolution-type schemes is $\alpha$-robust, i.e., the factor will not blowup while $\alpha\to 1^-$ with general nonuniform time steps even when rather general initial regularity condition is considered.