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arXiv · 1803.09879

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

Hong-lin Liao·William McLean·Jiwei Zhang
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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.

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Hong-lin Liao, William McLean, Jiwei Zhang. 2018-03-27. A discrete Gr\"{o}nwall inequality with application to numerical schemes for subdiffusion problems. https://doi.org/10.1137/16m1175742

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