  Power-law probability density function (PDF) plays a key role in both
subdiffusion and L\'{e}vy flights. However, sometimes because of the finite of
the lifespan of the particles or the boundedness of the physical space,
tempered power-law PDF seems to be a more physical choice and then the tempered
fractional operators appear; in fact, the tempered fractional operators can
also characterize the transitions among subdiffusion, normal diffusion, and
L\'{e}vy flights. This paper focuses on the quasi-compact schemes for space
tempered fractional diffusion equations, being much different from the ones for
pure fractional derivatives. By using the generation function of the matrix and
Weyl's theorem, the stability and convergence of the derived schemes are
strictly proved. Some numerical simulations are performed to testify the
effectiveness and numerical accuracy of the obtained schemes.
