A discrete Gr\"{o}nwall inequality with application to numerical schemes for subdiffusion problems

TL;DR

This paper introduces a discrete Grönwall inequality tailored for numerical schemes approximating the Caputo fractional derivative, enabling stability and convergence analysis for subdiffusion problems with nonuniform time steps.

Contribution

It extends existing Grönwall inequalities to nonuniform time steps and higher order schemes, improving analysis tools for fractional differential equations.

Findings

Established a fractional Grönwall inequality applicable to nonuniform time steps.

Demonstrated stability and convergence estimates for fractional reaction-subdiffusion schemes.

Extended analysis framework beyond the L1 approximation to higher order methods.

Abstract

We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose derivatives are singular at~$t=0$. The main result is a type of fractional Gr\"{o}nwall inequality and we illustrate its use by outlining some stability and convergence estimates of schemes for fractional reaction-subdiffusion problems. This approach extends earlier work that used the familiar L1 approximation to the Caputo fractional derivative, and will facilitate the analysis of higher order and linearized fast schemes.