  The continuous time random walk (CTRW) underlies many fundamental processes
in non-equilibrium statistical physics. When the jump length of CTRW obeys a
power-law distribution, its corresponding Fokker-Planck equation has space
fractional derivative, which characterizes L\'{e}vy flights. Sometimes the
infinite variance of L\'{e}vy flight discourages it as a physical approach;
exponentially tempering the power-law jump length of CTRW makes it more
`physical' and the tempered space fractional diffusion equation appears. This
paper provides the basic strategy of deriving the high order quasi-compact
discretizations for space fractional derivative and tempered space fractional
derivative. The fourth order quasi-compact discretization for space fractional
derivative is applied to solve space fractional diffusion equation and the
unconditional stability and convergence of the scheme are theoretically proved
and numerically verified. Furthermore, the tempered space fractional diffusion
equation is effectively solved by its counterpart of the fourth order
quasi-compact scheme; and the convergence orders are verified numerically.

Numerical Analysis