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

Local discontinuous Galerkin methods for the time tempered fractional diffusion equation