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A second-order numerical method for two-sided tempered fractional convection-diffusion equations is studied in this paper, both convection term and diffusion term are approximated by the tempered weighted and shifted Grünwald difference operators, the first time partial derivative is discretized by the Crank–Nicolson method, and then a class of second-order numerical schemes is derived. By means of matrix method, numerical schemes are proved to be unconditionally stable and convergent with order O τ 2 + h 2 . The validity of the proposed numerical scheme is verified by numerical experiments. [ABSTRACT FROM AUTHOR]