A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems

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 temporal derivatives are singular at t = 0. The main result is a type of fractional Grönwall 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. Key words. fractional subdiffusion equations, nonuniform time mesh, discrete Caputo derivative, discrete Grönwall inequality.