The unified theory of shifted convolution quadrature for fractional calculus

Abstract

The convolution quadrature theory is a systematic approach to analyse the approximation of the Riemann-Liouville fractional operator I α 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 θ to cover as many numerical schemes that approximate the operator I α with an integer convergence rate as possible. The constraint on the parameter θ 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.

Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes

“…The method adopted in this paper to discretize temporal direction is known as shifted fractional trapezoid rule (SFTR) first proposed in [13], which belongs to time-stepping methods. Historically, several groups of time-stepping methods have been devised such as the convolution quadrature (CQ) [7], the L type formulas (L1, L1-2, L2-1 σ ) [4,5,[14][15][16][17], the weighted and shifted Grünwald difference operators (WSGD) [11,18] and shifted convolution quadrature (SCQ) [6,33], and so on. Generally, since the subdiffusion problem (1.1) is characterized by the solution singularity at initial time [9,10], researchers have paid special attention on developing novel techniques such as by using nonuniform meshes [19,20,24] or adding correction terms [1,8,21,29,31] to restore the optimal convergence rate.…”