High dimensional conservative spatial distributed-order fractional diffusion equation is discretized by midpoint quadrature rule, Crank–Nicolson method, and a finite volume approximation, with alternating direction implicit scheme. The resulting scheme is shown to be consistent and unconditionally stable, hence convergent with order $3-\alpha$, where $\alpha$ is the maximum of the involving fractional orders. Moreover, if the initial condition and source term possess Tensor-Train format (TT-format) with low TT-ranks, the scheme can be solved in TT-format, such that higher dimensional cases can be considered. Perturbation analysis ensures that the accumulated errors due to data recompression do not affect the overall convergence order. Numerical examples with low TT-ranks initial conditions and source terms, and with dimensions up to 20 are tested.

通过中点正交法则，Crank–Nicolson方法和有限体积近似以及交替方向隐式格式离散高维保守空间分布阶分数阶扩散方程。结果表明方案是一致且无条件稳定的，因此与阶收敛$3-\alpha$， 在哪里 $\alpha$是涉及的分数阶的最大值。此外，如果初始条件和源项具有低TT等级的Tensor-Train格式（TT格式），则可以以TT格式解决该方案，从而可以考虑更高维的情况。扰动分析可确保由于数据重新压缩而导致的累积错误不会影响整体收敛顺序。测试了具有低TT等级初始条件和源项且尺寸最大为20的数值示例。