On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

TL;DR

This paper introduces a second-order implicit scheme for time-space fractional Bloch-Torrey equations, develops an all-at-once system, and proposes a bilateral preconditioning technique to efficiently solve it, with proven stability, convergence, and bounded condition number.

Contribution

The paper presents a novel second-order implicit difference scheme for TSFBTEs and a bilateral preconditioning method for the all-at-once system to enhance parallel-in-time solution efficiency.

Findings

The proposed scheme is stable and convergent.

The preconditioning technique accelerates Krylov solver convergence.

The condition number of the preconditioned matrix is uniformly bounded for certain fractional orders.

Abstract

Time-space fractional Bloch-Torrey equations (TSFBTEs) are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order implicit difference scheme for TSFBTEs by employing the recently proposed L2-type formula [A.~A.~Alikhanov, C.~Huang, Appl.~Math.~Comput.~(2021) 126545]. Then, we prove the stability and the convergence of the proposed scheme. Based on such a numerical scheme, an L2-type all-at-once system is derived. In order to solve this system in a parallel-in-time pattern, a bilateral preconditioning technique is designed to accelerate the convergence of Krylov subspace solvers according to the special structure of the coefficient matrix of the system. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant for the time fractional order $\alpha \in (0,0.3624)$. Numerical results are reported to show the efficiency of our method.