A nonuniform L2-1$_\sigma$/LDG method for the Caputo-Hadamard time-fractional convection-diffusion equation

In this paper, we consider the efficient numerical approach for the time-fractional convection-diffusion equation with Caputo-Hadamard derivative. This method uses the nonuniform L2-1$_\sigma$ formula for the time-fractional derivative and the local discontinuous Galerkin (LDG) method for the space approximation. In order to analyze the stability and convergence of the algorithm, a new discrete Gronwall inequality related to the discretized model with Caputo-Hadamard derivative is established. The result shows that the method has $\alpha$-robust, i.e., it remains valid when $\alpha\rightarrow 1^-$. Finally, the theoretical results are further verified by a numerical example.