In this article, two kinds of numerical algorithms are derived for the ultra-slow (or superslow) diffusion equation in one and two space dimensions, where the ultra-slow diffusion is characterized by the Caputo-Hadamard fractional derivative of order $\alpha \in (0,1)$. To describe the spatial interaction, the Riesz fractional derivative and the fractional Laplacian are used in one and two space dimensions, respectively. The Caputo-Hadamard derivative is discretized by two typical approximate formulae, i.e., L2-1$_{\sigma}$ and L1-2 methods. The spatial fractional derivatives are discretized by the 2-nd order finite difference methods. When L2-1$_{\sigma}$ discretization is used, the derived numerical scheme is unconditionally stable with error estimate $\mathcal{O}(\tau^{2}+h^{2})$ for all $\alpha \in (0, 1)$, in which $\tau$ and $h$ are temporal and spatial stepsizes, respectively. When L1-2 discretization is used, the derived numerical scheme is stable with error estimate $\mathcal{O}(\tau^{3-\alpha}+h^{2})$ for $\alpha \in (0, 0.3738)$. The illustrative examples displayed are in line with the theoretical analysis.

翻译：本文针对一维和二维空间中的超慢扩散方程，推导了两类数值算法。其中超慢扩散过程由阶数为α∈(0,1)的Caputo-Hadamard分数阶导数刻画。在一维和二维空间中，分别采用Riesz分数阶导数和分数阶Laplacian算子描述空间相互作用。Caputo-Hadamard导数通过两种典型近似公式（即L2-1σ方法和L1-2方法）进行离散，空间分数阶导数采用二阶有限差分方法离散。采用L2-1σ离散时，所导出的数值格式对所有α∈(0,1)均无条件稳定且具有误差估计O(τ²+h²)，其中τ和h分别为时间和空间步长；采用L1-2离散时，所导出的数值格式在α∈(0,0.3738)范围内稳定且具有误差估计O(τ³⁻ᵃ+h²)。文中的算例结果与理论分析一致。