Strongly Coupled Two-scale System with Nonlinear Dispersion: Weak Solvability and Numerical Simulation
Abstract
We investigate a two-scale system featuring an upscaled parabolic dispersion-reaction equation intimately linked to a family of elliptic cell problems. The system is strongly coupled through a dispersion tensor, which depends on the solutions to the cell problems, and via the cell problems themselves, where the solution of the parabolic problem interacts nonlinearly with the drift term. This particular mathematical structure is motivated by a rigorously derived upscaled reaction-diffusion-convection model that describes the evolution of a population of interacting particles pushed by a large drift through an array of periodically placed obstacles (i.e., through a regular porous medium). We prove the existence and uniqueness of weak solutions to our system by means of an iterative scheme, where particular care is needed to ensure the uniform positivity of the dispersion tensor. Additionally, we use finite element-based approximations for the same iteration scheme to perform multiple simulation studies. Finally, we highlight how the choice of micro-geometry (building the regular porous medium) and of the nonlinear drift coupling affects the macroscopic dispersion of particles.
Cite
@article{arxiv.2311.12251,
title = {Strongly Coupled Two-scale System with Nonlinear Dispersion: Weak Solvability and Numerical Simulation},
author = {Vishnu Raveendran and Surendra Nepal and Rainey Lyons and Michael Eden and Adrian Muntean},
journal= {arXiv preprint arXiv:2311.12251},
year = {2025}
}