In this paper, we develop an efficient spectral Galerkin method for the three-dimensional (3D) multi-term time-space fractional diffusion equation. Based on the L2-1_σ formula for time stepping and the Legendre-Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed and the stability and convergence analyses are rigorously established. The results show that the fully discrete scheme is unconditionally stable and has second-order accuracy in time and optimal error estimation in space. In addition, we give the detailed implementation and apply the alternating-direction implicit (ADI) method to reduce the computational complexity. Furthermore, numerical experiments are presented to confirm the theoretical claims. As an application of the proposed method, the fractional Bloch-Torrey model is also solved.

在本文中，我们为三维（3D）多维时空分数阶扩散方程开发了一种有效的光谱Galerkin方法。基于所述大号2-1 _σ通过时间步长公式和空间离散化的Legendre-Galerkin谱方法，构造了一个完全离散的数值格式，并严格建立了稳定性和收敛性分析。结果表明，完全离散的方案是无条件稳定的，在时间上具有二阶精度，并且在空间上具有最佳误差估计。此外，我们给出了详细的实现方式，并应用了交替方向隐式（ADI）方法来降低计算复杂度。此外，提出了数值实验以确认理论要求。作为所提出方法的应用，还解决了分数Bloch-Torrey模型。