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

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