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

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.