Analysis and numerical implementation of high-order method for distributed-order diffusion model

This work proposes a high-order discontinuous Galerkin (DG) formulation employing generalized alternating numerical fluxes for approximating solutions to the distributed-order diffusion equation. Such models often arise in ultraslow diffusion, where solutions decay only logarithmically as $t\to \mathrm{\infty }$. Utilizing the Grünwald–Letnikov scheme and the DG method, we construct a fully discrete numerical algorithm. Using a rigorous induction argument, we prove unconditional stability and convergence of the proposed scheme. A comprehensive set of computational experiments is presented to validate the efficacy and performance of the method.