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.

翻译：一些研究人员开发了时间- 空间分块布洛- 托雷方程式( TSFBTEs ), 以调查扩散和分序动态之间的关系。 在本文中, 我们首先通过使用最近提议的 L2 型公式[A. ~ Alikhanov, C~Huanganov, Appl. ~Math. ~Comput. ~ (2021) 126.545], 来为 TSFBTEs 提出二级隐含差异方案。 然后, 我们证明拟议方案的稳定性和趋同性。 根据这样一个数字方案, 将产生一个L2型全自动系统。 为了以平行时间模式解决这个系统, 我们设计了一种双边先决条件技术, 以加速Krylov 子空间解答器与系统系数矩阵的特殊结构的趋同。 我们从理论上表明, 先决条件矩阵的条件号由一个恒定的零时分序 $\ alpha \ in ( 0, 0.3624) 。 据报告, 数字结果显示我们的方法的效率。