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High order discretization schemes play more important role in fractionaloperators than classical ones. This is because usually for classicalderivatives the stencil for high order discretization schemes is wider than loworder ones; but for fractional operators the stencils for high order schemesand low order ones are the same. Then using high order schemes to solvefractional equations leads to almost the same computational cost with firstorder schemes but the accuracy is greatly improved. Using the fractional linearmultistep 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 thestability issue the obtained scheme can not be directly applied to the spacefractional operator with $\alpha \in(1,2)$ for time dependent problem. Byweighting and shifting Lubich's 2nd order discretization scheme, in [Chen &Deng, arXiv:1304.7425] we derive a series of effective high orderdiscretizations for space fractional derivative, called WSLD opeartors there.As the sequel of the previous work, we further provide new high order schemesfor space fractional derivatives by weighting and shifting Lubich's 3rd and 4thorder discretizations. In particular, we prove that the obtained 4th orderapproximations are effective for space fractional derivatives. And thecorresponding schemes are used to solve the space fractional diffusion equationwith variable coefficients.

WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative