In this paper, we have proposed a numerical approach based on generalized alternating numerical fluxes to solve the multi-term fractional reaction-diffusion equation. This type of equation frequently arises in the mathematical modeling of ultra-slow diffusion phenomena observed in various physical problems. These phenomena are characterized by solutions that exhibit logarithmic decay as time $t$ approaches infinity. For spatial discretization, we employed the discontinuous Galerkin method with generalized alternating numerical fluxes. Temporal discretization was handled using the finite difference method. To ensure the robustness of the proposed scheme, we rigorously established its unconditional stability through mathematical induction. Finally, we conducted a series of comprehensive numerical experiments to validate the accuracy and efficiency of the scheme, demonstrating its potential for practical applications.

In this work, we present a high-order discontinuous Galerkin (DG) method with generalized alternating numerical fluxes to solve the variable-order (VO) fractional mobile-immobile advection-dispersion equation. This equation models complex transport phenomena where the order of differentiation varies with time, providing a more accurate representation of anomalous diffusion in heterogeneous media. For spatial and temporal discretization, the method employs the DG scheme and a finite difference method, respectively. Rigorous analysis confirms that the numerical scheme is unconditionally stable and convergent. Finally, numerical experiments are conducted to validate the theoretical results and illustrate the accuracy and efficiency of the scheme.