  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.
