In this paper, we consider the numerical approximation of the space and time fractional Bloch-Torrey equations. A fully discrete spectral scheme based on a finite difference method in the time direction and a Galerkin-Legendre spectral method in the space direction is developed. In order to reduce the amount of computation, an alternating direction implicit (ADI) spectral scheme is proposed. Then the stability and convergence analysis are rigorously established. Finally, numerical results are presented to support our theoretical analysis.

中文翻译：

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二维时空分数Bloch-Torrey方程的频谱方法

在本文中，我们考虑了时空分数Bloch-Torrey方程的数值逼近。提出了一种基于时间方向上的有限差分法和空间方向上的Galerkin-Legendre光谱法的全离散光谱方案。为了减少计算量，提出了一种交替方向隐式（ADI）频谱方案。然后严格建立稳定性和收敛性分析。最后，给出数值结果以支持我们的理论分析。