  In this paper, we propose third-order semi-discretized schemes in space based
on the tempered weighted and shifted Gr\"unwald difference (tempered-WSGD)
operators for the tempered fractional diffusion equation. We also show
stability and convergence analysis for the fully discrete scheme based a
Crank--Nicolson scheme in time. A third-order scheme for the tempered
Black--Scholes equation is also proposed and tested numerically. Some numerical
experiments are carried out to confirm accuracy and effectiveness of these
proposed methods.
