An Upwind Interior Penalty DG Scheme for Solute Transport in 2D Variable-Order Mobile–Immobile Model

This paper develops and rigorously analyzes a fully discrete upwind interior penalty discontinuous Galerkin (IPDG) scheme for simulating solute transport in two-dimensional variable-order fractional mobile–immobile media. The temporal variable-order Caputo derivative is discretized via a Grünwald–Letnikov approximation in conjunction with a first-order backward difference, while the spatial discretization employs an IPDG method featuring an upwind numerical flux for the convection term and a penalty formulation for the diffusion operator. Under the physically relevant assumption of a divergence-free velocity field, we establish the unconditional stability of the proposed scheme. A comprehensive error analysis in the L2 norm yields a convergence rate of O(Δt+hmin(k+1,s)−χ−1/2), explicitly linking the polynomial degree k, solution regularity s, and the penalty variant χ. Numerical experiments in two dimensions are conducted to verify the accuracy and robustness of the proposed scheme in simulating anomalous transport phenomena in subsurface environments.