An efficient numerical method for a time-fractional diffusion equation

TL;DR

This paper introduces an efficient second-order numerical method for solving reaction-diffusion equations with Caputo time derivatives, improving convergence rates over existing schemes and validated by numerical experiments.

Contribution

The paper proposes a novel integral discretization scheme on graded meshes with a solution decomposition, achieving second-order convergence for time-fractional diffusion equations.

Findings

Second-order convergence of the proposed scheme

Enhanced accuracy over L1 schemes

Numerical results confirm theoretical analysis

Abstract

A reaction-diffusion problem with a Caputo time derivative is considered. An integral discretization scheme on a graded mesh along with a decomposition of the exact solution is proposed. The truncation error estimate of the discretization scheme is derived by using the remainder formula of the linear interpolation and some inequality estimate techniques. It is proved that the scheme is second-order convergent by applying a difference analogue of Gronwall's inequality, which exhibits an enhancement in the convergence rate compared with the L1 schemes. Numerical experiments are presented to support the theoretical result.