An entropy-stabilized mean-mapped discontinuous Galerkin method for bound-preserving flow and transport in porous media

We develop an entropy-stabilized mean-mapped discontinuous Galerkin (DG) method for bound-preserving flow and transport in porous media. The method targets the element averages, which are the locally conserved quantities in a DG discretization and represent physically meaningful masses, concentrations, or saturations. Instead of computing a possibly inadmissible average and repairing it a posteriori, we separate the element average from the higher-order zero-mean DG modes and represent each cell average through a scalar entropy variable, u ̄ T = ( H ′ ) − 1 ( θ T ) , where H : ( m , M ) → R is strictly convex and H ′ maps the admissible interval ( m , M ) bijectively onto R . This parametrization enforces the prescribed bounds on the conservative cell averages by construction: the computed averages are admissible unknowns of the nonlinear discrete problem, not quantities repaired by clipping, projection, or a posteriori correction. The zero-mean DG modes are not mapped and remain available for high-order approximation. The scheme therefore supplies the admissible cell averages required by standard conservative mean-preserving limiters, which may then be applied, if desired, to enforce pointwise bounds on the full DG polynomial. To obtain a solvable and robust nonlinear formulation, we introduce a conservative stabilization acting only on neighboring entropy variables. The stabilization has the form of a graph-viscosity jump penalty on the cell-average equations. It contributes antisymmetric fluxes across interior faces and therefore preserves the local conservation of the stabilized cell-average balance. When scaled as a high-order perturbation, the stabilization controls the mapped averages without degrading the formal accuracy of the raw DG method in smooth regions. For a steady model problem, we prove existence of a finite-entropy discrete solution, local conservation of the stabilized cell-average balance, and conditional preservation of the convergence order of the underlying DG approximation under high-order scaling of the entropy-variable stabilization. Numerical experiments for steady convection-dominated transport, non-simplicial meshes, and transient miscible displacement in homogeneous and heterogeneous porous media show that the proposed method removes observed cell-average undershoots and overshoots while maintaining local conservation and retaining the high-order structure of the DG approximation.