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.

翻译：我们考虑一阶广义卷积求积（gCQ）在逼近分数阶积分及相关分数阶扩散方程中的应用。广义卷积求积是Lubich卷积求积（CQ）的一种推广，允许使用变步长。本文分析了gCQ在分数阶积分中的应用，重点关注低正则性情形。众所周知，在此情形下，原始CQ方法在奇点附近会出现降阶现象。现有gCQ理论并未涵盖这种情况。本文采用数值逼近及其误差的另一种表达式，从而显著放宽了应用gCQ方法所需的正则性条件。特别地，我们能够消除原始gCQ推导中所需的先验正则化步骤。在比文献中已有结果弱得多的正则性要求下，我们证明了在一般时间网格上具有一阶收敛性。同时证明，对于分级时间网格（该网格根据分数阶积分的阶次和数据正则性在奇点附近适当加密），可以实现一致一阶收敛。进而，我们研究了在线性分数阶扩散方程应用中实现完整收敛阶的方法。gCQ方法的一个重要优势在于能够实现快速且低内存占用的算法实现。我们概述了该算法的实现流程，并通过多个数值实验验证了理论结果。