Sharp-diffuse interfaces model with cross-diffusion in a free aquifer : numerical analysis of a finite volume scheme
Ahmed Ait Hammou Oulhaj, Éloïse Comte, Moussa Mory Moussa Mory Diedhiou, David Maltese · HAL (Le Centre pour la Communication Scientifique Directe) · 2024
Sharp-diffuse interface is a new model for seawater intrusion problem which have the efficiency of the sharp interface approach and the physical realism of the diffuse interface. The introduction of the diffuse interface as a phase field in the new model makes it non-degenerate. Three-dimensional problem is reduced to a two-dimensional model involving a strongly coupled system of PDE of parabolic type describing the evolution of the depths of the interface between salt-and freshwater and the water table. We propose and analyze a finite volume scheme based on two-point flux approximation with upwind mobilities. The scheme preserves at the discrete level the main features of the continuous problem, namely the nonnegativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. Based on these nonlinear stability results, we show that the scheme converges towards a weak solution to the problem. Numerical results are provided toillustrate the behavior of the model and of the scheme.