A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems

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.