Local discontinuous Galerkin methods for the time tempered fractional diffusion equation

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.