Analysis of Local Discontinuous Galerkin Method for the Variable-order Subdiffusion Equation with the Caputo–Hadamard Derivative

Abstract

In this article, we investigate a high-order numerical method for the variable-order (VO) subdiffusion equation with the Caputo–Hadamard derivative. The temporal variable and the spatial variable are discretized by the finite difference method and the local discontinuous Galerkin method, respectively. Furthermore, for all variable-order $\alpha(t) \in (0,1)$, the stability and the optimal error estimates are proved for the presented scheme. Finally, several numerical tests are given to demonstrate optimal rates of convergence and show the efficiency of the method.