  In this article, we consider discrete schemes for a fractional diffusion
equation involving a tempered fractional derivative in time. We present a
semi-discrete scheme by using the local discontinuous Galerkin (LDG)
discretization in the spatial variables. We prove that the semi-discrete scheme
is unconditionally stable in $L^2$ norm and convergence with optimal
convergence rate $\mathcal{O}(h^{k+1})$. We develop a class of fully discrete
LDG schemes by combining the weighted and shifted Lubich difference operators
with respect to the time variable, and establish the error estimates. Finally,
numerical experiments are presented to verify the theoretical results.
