A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems
For researchers analyzing numerical methods for fractional differential equations, this provides a general tool for stability and convergence proofs, but it is an incremental extension of existing Grönwall-type inequalities.
The authors prove a discrete fractional Grönwall inequality for numerical approximations of Caputo fractional derivatives with nonuniform time steps, enabling stability and convergence analysis of schemes for subdiffusion problems. The result extends prior work on the L1 approximation to higher-order and fast linearized schemes.
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 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.