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The continuous time random walk (CTRW) underlies many fundamental processesin non-equilibrium statistical physics. When the jump length of CTRW obeys apower-law distribution, its corresponding Fokker-Planck equation has spacefractional derivative, which characterizes L\'{e}vy flights. Sometimes theinfinite 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. Thispaper provides the basic strategy of deriving the high order quasi-compactdiscretizations for space fractional derivative and tempered space fractionalderivative. The fourth order quasi-compact discretization for space fractionalderivative is applied to solve space fractional diffusion equation and theunconditional stability and convergence of the scheme are theoretically provedand numerically verified. Furthermore, the tempered space fractional diffusionequation is effectively solved by its counterpart of the fourth orderquasi-compact scheme; and the convergence orders are verified numerically.

Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations