Numerical Exploration of Nonlinear Dispersion Effects via a Strongly Coupled Two-scale System

Abstract: The effective, fast transport of matter through porous media is often characterized by complex dispersion effects. To describe in mathematical terms such situations, instead of a simple macroscopic equation (as in the classical Darcy's law), one may need to consider two-scale boundary-value problems with full coupling between the scales where the macroscopic transport depends non-linearly on local (i.e. microscopic) drift interactions, which are again influenced by local concentrations. Such two-scale problems are computationally very expensive as numerous elliptic partial differential equations (cell problems) have to constantly be recomputed. In this work, we investigate such an effective two-scale model involving a suitable nonlinear dispersion term and explore numerically the behavior of its weak solutions. We introduce two distinct numerical schemes dealing with the same non-linear scale-coupling: (i) a Picard-type iteration and (ii) a time discretization decoupling. In addition, we propose a precomputing strategy where the calculations of cell problems are pushed into an offline phase. Our approach works for both schemes and significantly reduces computation times. We prove that the proposed precomputing strategy converges to the exact solution. Finally, we test our schemes via several numerical experiments that illustrate dispersion effects introduced by specific choices of microstructure and model ingredients.