  The tempered fractional diffusion equation could be recognized as the
generalization of the classic fractional diffusion equation that the truncation
effects are included in the bounded domains. This paper focuses on designing
the high order fully discrete local discontinuous Galerkin (LDG) method based
on the generalized alternating numerical fluxes for the tempered fractional
diffusion equation. From a practical point of view, the generalized alternating
numerical flux which is different from the purely alternating numerical flux
has a broader range of applications. We first design an efficient finite
difference scheme to approximate the tempered fractional derivatives and then a
fully discrete LDG method for the tempered fractional diffusion equation. We
prove that the scheme is unconditionally stable and convergent with the order
$O(h^{k+1}+\tau^{2-\alpha})$, where $h, \tau$ and $k$ are the step size in
space, time and the degree of piecewise polynomials, respectively. Finally
numerical experimets are performed to show the effectiveness and testify the
accuracy of the method.
