  Based on the superconvergent approximation at some point (depending on the
fractional order $\alpha$, but not belonging to the mesh points) for
Gr\"{u}nwald discretization to fractional derivative, we develop a series of
high order quasi-compact schemes for space fractional diffusion equations.
Because of the quasi-compactness of the derived schemes, no points beyond the
domain are used for all the high order schemes including second order, third
order, fourth order, and even higher order schemes; moreover, the algebraic
equations for all the high order schemes have the completely same matrix
structure. The stability and convergence analysis for some typical schemes are
made; the techniques of treating the nonhomogeneous boundary conditions are
introduced; and extensive numerical experiments are performed to confirm the
theoretical analysis or verify the convergence orders.
