# The unified theory of shifted convolution quadrature for fractional calculus

## TL;DR

This paper introduces the shifted convolution quadrature ($SCQ$) theory, extending existing methods for approximating fractional operators by incorporating a shift parameter, analyzing superconvergence, and validating with numerical tests.

## Contribution

The paper develops a generalized $SCQ$ theory with a shifted parameter, broadening the scope of convolution quadrature methods for fractional calculus and analyzing superconvergence phenomena.

## Key findings

- Superconvergence observed in some schemes.
- New formulas designed for stability and convergence analysis.
- Numerical tests confirm theoretical predictions.

## Abstract

The convolution quadrature theory is a systematic approach to analyse the approximation of the Riemann-Liouville fractional operator $I^{\alpha}$ at node $x_{n}$. In this paper, we develop the shifted convolution quadrature ($SCQ$) theory which generalizes the theory of convolution quadrature by introducing a shifted parameter $\theta$ to cover as many numerical schemes that approximate the operator $I^{\alpha}$ with an integer convergence rate as possible. The constraint on the parameter $\theta$ is discussed in detail and the phenomenon of superconvergence for some schemes is examined from a new perspective. For some technique purposes when analysing the stability or convergence estimates of a method applied to PDEs, we design some novel formulas with desired properties under the framework of the $SCQ$. Finally, we conduct some numerical tests with nonsmooth solutions to further confirm our theory.