Efficient numerical schemes for variable-order mobile-immobile advection-dispersion equation

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.