  High order discretization schemes play more important role in fractional
operators than classical ones. This is because usually for classical
derivatives the stencil for high order discretization schemes is wider than low
order ones; but for fractional operators the stencils for high order schemes
and low order ones are the same. Then using high order schemes to solve
fractional equations leads to almost the same computational cost with first
order schemes but the accuracy is greatly improved. Using the fractional linear
multistep methods, Lubich obtains the $\nu$-th order ($\nu\leq 6$)
approximations of the $\alpha$-th derivative ($\alpha>0$) or integral
($\alpha<0) [Lubich, SIAM J. Math. Anal., 17, 704-719, 1986], because of the
stability issue the obtained scheme can not be directly applied to the space
fractional operator with $\alpha \in(1,2)$ for time dependent problem. By
weighting and shifting Lubich's 2nd order discretization scheme, in [Chen &
Deng, arXiv:1304.7425] we derive a series of effective high order
discretizations for space fractional derivative, called WSLD opeartors there.
As the sequel of the previous work, we further provide new high order schemes
for space fractional derivatives by weighting and shifting Lubich's 3rd and 4th
order discretizations. In particular, we prove that the obtained 4th order
approximations are effective for space fractional derivatives. And the
corresponding schemes are used to solve the space fractional diffusion equation
with variable coefficients.
