  Because of the nonlocal properties of fractional operators, higher order
schemes play more important role in discretizing fractional derivatives than
classical ones. The striking feature is that higher order schemes of fractional
derivatives can keep the same computation cost with first-order schemes but
greatly improve the accuracy. Nowadays, there are already two types of second
order discretization schemes for space fractional derivatives: the first type
is given and discussed in [Sousa & Li, arXiv:1109.2345; Chen & Deng,
arXiv:1304.3788; Chen et al., Appl. Numer. Math., 70, 22-41]; and the second
type is a class of schemes presented in [Tian et al., arXiv:1201.5949]. The
core object of this paper is to derive a class of fourth order approximations,
called the weighted and shifted Lubich difference (WSLD) operators, for space
fractional derivatives. Then we use the derived schemes to solve the space
fractional diffusion equation with variable coefficients in one-dimensional and
two-dimensional cases. And the unconditional stability and the convergence with
the global truncation error $\mathcal{O}(\tau^2+h^4)$ are theoretically proved
and numerically verified.
