Analysis of the L1 scheme for fractional wave equations with nonsmooth data

TL;DR

This paper investigates the stability and accuracy of the L1 scheme for fractional wave equations with nonsmooth data, proposing modifications and verifying results through numerical experiments.

Contribution

It introduces a new stability estimate, derives improved accuracy results, and proposes a modified L1 scheme for better handling nonsmooth initial data.

Findings

Established stability and accuracy of the L1 scheme for nonsmooth data

Proposed a modified L1 scheme with enhanced stability and accuracy

Numerical experiments confirm theoretical predictions

Abstract

This paper analyzes the well-known L1 scheme for fractional wave equations with nonsmooth data. A new stability estimate is obtained, and the temporal accuracy $ \mathcal O(\tau^{3-\alpha}) $ is derived for the nonsmooth initial data. In addition, a modified L1 scheme is proposed, and stability and temporal accuracy $ \mathcal O(\tau^2) $ are derived for this scheme with nonsmooth initial data. The convergence of the two schemes in the inhomogeneous case is also established. Finally, numerical experiments are performed to verify the theoretical results.