Fast iterative method with a second order implicit difference scheme for time-space fractional convection-diffusion equations

TL;DR

This paper introduces a second-order accurate implicit difference scheme for solving time-space fractional convection-diffusion equations, along with fast Krylov subspace solvers that significantly reduce computational cost and memory usage.

Contribution

It develops a practical, stable, and convergent second-order implicit difference method and efficient Krylov-based solvers for large-scale fractional PDEs.

Findings

Second-order convergence in time and space.

Reduced memory from O(N^2) to O(N).

Reduced computational complexity from O(N^3) to O(N log N).

Abstract

In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Gr\"{u}nwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $O(N^3)$ to $O(N\log N)$ in each iterative step, where $N$ is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.