Fast algorithms for high-dimensional variable-order space-time fractional diffusion equations

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 Grunwald 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.