Time-space fractional Bloch-Torrey equations are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order scheme for this equation 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 this scheme. Based on such the numerical scheme, a 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 according to the special structure 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.

中文翻译：

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关于由时空分数布洛赫-托雷方程产生的 L2 型一次性系统的双边预处理

一些研究人员开发了时空分数 Bloch-Torrey 方程来研究扩散和分数阶动力学之间的关系。在本文中，我们首先通过采用最近提出的 L2 型公式 [A.~A.~Alikhanov, C.~Huang, Appl.~Math.~Comput.~(2021) 126545]。然后，我们证明了该方案的稳定性和收敛性。基于这样的数值方案，推导出L2型all-at-once系统。为了以时间并行方式求解该系统，根据系统的特殊结构设计了双边预处理技术。我们理论上表明，预处理矩阵的条件数由时间分数阶 $\alpha\in (0,0.3624)$ 的常数统一限定。