Preconditioned iterative methods for space-time fractional advection-diffusion equations

TL;DR

This paper develops efficient numerical methods, including preconditioned iterative solvers, for solving space-time fractional advection-diffusion equations, demonstrating their stability, efficiency, and applicability to variable coefficients.

Contribution

It introduces a practical implicit scheme and preconditioned iterative solvers with FFT acceleration for space-time fractional advection-diffusion equations, enhancing computational efficiency.

Findings

The implicit method is stable and consistent.

Preconditioned GMRES and CGNR methods significantly reduce computational cost.

Numerical experiments confirm the efficiency even with variable coefficients.

Abstract

In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of space-time fractional advection-diffusion equations. To start with, an implicit method based on two-sided Gr\"unwald formulae is proposed with a discussion of the stability and consistency. Then, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual ({preconditioned} CGNR) method, with an easily constructed preconditioner, are developed. Importantly, because the resulting systems are Topelitz-like, the fast Fourier transform can be applied to significantly reduce the computational cost. Numerical experiments are implemented to show the efficiency of our preconditioner, even with cases of variable coefficients.