An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data

TL;DR

This paper revisits the L1 scheme for the subdiffusion equation, establishing an $O( au)$ convergence rate for both smooth and nonsmooth data, and extends analysis to more general operators with supporting numerical experiments.

Contribution

The paper provides a new error analysis for the L1 scheme, achieving an $O( au)$ rate without requiring solution smoothness, and applies it to more general fractional diffusion equations.

Findings

Error estimate of $O( au)$ for nonsmooth data

Applicability to general sectorial operators

Numerical validation of theoretical results

Abstract

The subdiffusion equation with a Caputo fractional derivative of order $\alpha\in(0,1)$ in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for discretizing the Caputo fractional derivative in time. The scheme was analyzed earlier independently by Lin and Xu (2007) and Sun and Wu (2006), and an $O(\tau^{2-\alpha})$ convergence rate was established, under the assumption that the solution is twice continuously differentiable in time. However, in view of the smoothing property of the subdiffusion equation, this regularity condition is restrictive, since it does not hold even for the homogeneous problem with a smooth initial data. In this work, we revisit the error analysis of the scheme, and establish an $O(\tau)$ convergence rate for both smooth and nonsmooth initial data. The analysis is valid for more general sectorial operators. In particular, the L1 scheme is applied to one-dimensional space-time fractional diffusion equations, which involves also a Riemann-Liouville derivative of order $\beta\in(3/2,2)$ in space, and error estimates are provided for the fully discrete scheme. Numerical experiments are provided to verify the sharpness of the error estimates, and robustness of the scheme with respect to data regularity.