# Local discontinuous Galerkin methods for the time tempered fractional diffusion equation

## TL;DR

This paper develops and analyzes local discontinuous Galerkin methods for solving time-tempered fractional diffusion equations, proving stability, convergence, and validating results with numerical experiments.

## Contribution

It introduces a semi-discrete LDG scheme for tempered fractional diffusion and extends it to fully discrete schemes with error estimates, a novel approach for this class of equations.

## Key findings

- Semi-discrete LDG scheme is unconditionally stable in L2 norm.
- Optimal convergence rate of O(h^{k+1}) is achieved.
- Numerical experiments confirm theoretical error estimates.

## Abstract

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.