Generalized convolution quadrature for the fractional integral and fractional diffusion equations

We consider the application of the generalized Convolution Quadrature (gCQ) of the first order to approximate fractional integrals and associated fractional diffusion equations. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) which allows for variable steps. In this paper we analyze the application of the gCQ to fractional integrals, with a focus on the low regularity case. It is well known that in this situation the original CQ presents an order reduction close to the singularity. The available theory for the gCQ does not cover this situation. Here we use a different expression for the numerical approximation and the associated error, which allows us to significantly relax the regularity requirements for the application of the gCQ method. In particular we are able to eliminate the a priori regularization step required in the original derivation of the gCQ. We show first order of convergence for a general time mesh under much weaker regularity requirements than previous results in the literature. We also prove that uniform first order convergence is achievable for a graded time mesh, which is appropriately refined close to the singularity, according to the order of the fractional integral and the regularity of the data. Then we study how to obtain full order of convergence for the application to linear fractional diffusion equations. An important advantage of the gCQ method is that it allows for a fast and memory reduced implementation. We outline how this algorithm can be implemented and illustrate our theoretical results with several numerical experiments.