Construction of high-order robust theta-methods with applications in anomalous models

TL;DR

This paper introduces a new high-order theta-method with correction terms for fractional calculus operators, improving accuracy and robustness in solving subdiffusion problems with nonsmooth data.

Contribution

A novel theta-scheme with correction terms is developed for fractional subdiffusion equations, enhancing stability and accuracy for small fractional orders and nonsmooth initial data.

Findings

Achieves high-order accuracy in fractional calculus operators

Robust performance for very small fractional orders

Numerical tests confirm theoretical error estimates

Abstract

A general conversion strategy by involving a shifted parameter $\theta$ is proposed to construct high-order accuracy difference formulas for fractional calculus operators. By converting the second-order backward difference formula with such strategy, a novel $\theta$-scheme with correction terms is developed for the subdiffusion problem with nonsmooth data, which is robust even for very small $\alpha$ and can resolve the initial singularity.The optimal error estimates are carried out with essential arguments and are verified by numerical tests.