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The unified theory of shifted convolution quadrature for fractional calculus
The convolution quadrature theory is a systematic approach to analyse the
approximation of the Riemann-Liouville fractional operator at node
. In this paper, we develop the shifted convolution quadrature ()
theory which generalizes the theory of convolution quadrature by introducing a
shifted parameter to cover as many numerical schemes that approximate
the operator with an integer convergence rate as possible. The
constraint on the parameter 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 . Finally, we conduct some numerical
tests with nonsmooth solutions to further confirm our theory.Comment: 21 pages, 15 figure