Finite difference method as a popular numerical method has been widely used to solve fractional diffusion equations. In the general spatial error analyses, an assumption $u\in C^{4}(\bar{\Omega})$ is needed to preserve $\mathcal{O}(h^{2})$ convergence when using central finite difference scheme to solve fractional sub-diffusion equation with Laplace operator, but this assumption is somewhat strong, where $u$ is the exact solution and $h$ is the mesh size. In this paper, a novel analysis technique is proposed to show that the spatial convergence rate can reach $\mathcal{O}(h^{\min(\sigma+\frac{1}{2}-\epsilon,2)})$ in both $l^{2}$-norm and $l^{\infty}$-norm in one-dimensional domain when the initial value and source term are both in $\hat{H}^{\sigma}(\Omega)$ but without any regularity assumption on the exact solution, where $\sigma\geq 0$ and $\epsilon>0$ being arbitrarily small. After making slight modifications on the scheme, acting on the initial value and source term, the spatial convergence rate can be improved to $\mathcal{O}(h^{2})$ in $l^{2}$-norm and $\mathcal{O}(h^{\min(\sigma+\frac{3}{2}-\epsilon,2)})$ in $l^{\infty}$-norm. It's worth mentioning that our spatial error analysis is applicable to high dimensional cube domain by using the properties of tensor product. Moreover, two kinds of averaged schemes are provided to approximate the Riemann--Liouville fractional derivative, and $\mathcal{O}(\tau^{2})$ convergence is obtained for all $\alpha\in(0,1)$. Finally, some numerical experiments verify the effectiveness of the built theory.

翻译：有限差分法作为一种流行的数值方法，已被广泛用于求解分数阶扩散方程。在常规空间误差分析中，使用中心有限差分格式求解带拉普拉斯算子的分数阶次扩散方程时，需假设 $u\in C^{4}(\bar{\Omega})$ 以保证 $\mathcal{O}(h^{2})$ 收敛性，其中 $u$ 为精确解，$h$ 为网格步长，但这一假设较强。本文提出一种新颖的分析技巧，证明当初始值和源项均属于 $\hat{H}^{\sigma}(\Omega)$ 但不对精确解做任何正则性假定时，在一维区域上空间收敛率在 $l^{2}$-范数和 $l^{\infty}$-范数下均可达到 $\mathcal{O}(h^{\min(\sigma+\frac{1}{2}-\epsilon,2)})$，其中 $\sigma\geq 0$，$\epsilon>0$ 可任意小。通过对格式进行轻微修改（作用于初始值和源项），空间收敛率在 $l^{2}$-范数下可提升至 $\mathcal{O}(h^{2})$，在 $l^{\infty}$-范数下可提升至 $\mathcal{O}(h^{\min(\sigma+\frac{3}{2}-\epsilon,2)})$。值得指出的是，利用张量积性质，我们的空间误差分析可推广至高维立方体区域。此外，本文提供两种平均格式来近似Riemann-Liouville分数阶导数，并对所有 $\alpha\in(0,1)$ 实现 $\mathcal{O}(\tau^{2})$ 收敛。最后，数值实验验证了所建理论的有效性。