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

TL;DR

This paper develops correction techniques and fast algorithms for the shifted fractional trapezoidal rule to efficiently solve subdiffusion problems with nonsmooth solutions, achieving optimal accuracy and reduced computational cost.

Contribution

It introduces robust correction methods and fast algorithms for the SFTR, improving efficiency and accuracy in solving subdiffusion problems with nonsmooth initial data.

Findings

Fast algorithms reduce computational complexity to O(N log N)

Crank-Nicolson scheme restores optimal convergence without initial corrections for smooth data

Numerical tests confirm theoretical error estimates and efficiency

Abstract

This article devotes to developing robust but simple correction techniques and efficient algorithms for a class of second-order time stepping methods, namely the shifted fractional trapezoidal rule (SFTR), for subdiffusion problems to resolve the initial singularity and nonlocality. The stability analysis and sharp error estimates in terms of the smoothness of the initial data and source term are presented. As a byproduct in numerical tests, we find amazingly that the Crank-Nicolson scheme ($\theta=\frac{1}{2}$) without initial corrections can restore the optimal convergence rate for the subdiffusion problem with smooth initial data and source terms. To deal with the nonlocality, fast algorithms are considered to reduce the computational cost from $O(N^2)$ to $O(N \log N)$ and save the memory storage from $O(N)$ to $O(\log N)$, where $N$ denotes the number of time levels. Numerical tests are performed to verify the sharpness of the theoretical results and confirm the efficiency and accuracy of initial corrections and the fast algorithms.