High-dimensional variable-order space-time fractional diffusion equations have numerous real-world applications and have attracted a lot of attention in recent years. Solving this kind of equation is difficult because of the nonlocal property of time evolution. In this paper, we consider fast algorithms for high-dimensional variable-order space-time fractional diffusion equations. We propose an implicit discretization scheme based on Grünwald formula and discuss its stability and convergence. Two preconditioning strategies based on the structure of the coefficient matrix are proposed to reduce the computational costs. Furthermore, we design two low-rank tensor minimal energy (LTME) algorithms for multidimensional problems to solve relevant huge linear systems. Numerical examples are tested to show the effectiveness of the proposed methods.

高维可变阶时空分数扩散方程在现实世界中有许多应用，近年来引起了很多关注。由于时间演化的非局限性，因此很难求解这类方程。在本文中，我们考虑了高维可变阶时空分数阶扩散方程的快速算法。我们提出了一种基于Grünwald公式的隐式离散化方案，并讨论了其稳定性和收敛性。提出了两种基于系数矩阵结构的预处理策略，以减少计算量。此外，我们针对多维问题设计了两种低秩张量最小能量（LTME）算法，以解决相关的巨大线性系统。通过数值算例验证了所提方法的有效性。