Modèles et méthodes numériques pour des problèmes de transports multi-échelles
Laurent Navoret · HAL (Le Centre pour la Communication Scientifique Directe) · 2024
This manuscript presents different contributions in the mathematical modeling and the development of numerical methods for multiscale transport problems, arising in particular in plasma physics and collective cell dynamics.In the first chapter, we present a few reduction methods for kinetic transport equations. Performing a finite element semi-discretization in velocity leads to a linear hyperbolic model with good conservation properties. Fluid models with entropic or neural network closures have been studied to obtain even smaller reduced models with only physical quantities as reduced macroscopic variables. Reduced order modeling techniques are also proposed to construct reducedmacroscopic variables specifically adapted to a given set of parametric solutions.The second chapter deals with two numerical issues concerning the discontinuous Galerkin and its semi-Lagrangian version, namely the recurrence numerical artifact and the approximate well-balanced property by basis enrichment.The third chapter presents two implicit numerical methods for solving hyperbolic systems with low-computational cost thanks to relaxation techniques. The first method relies on the vectorial kinetic relaxation scheme, whose transport steps are solved using an implicit solver with explicitcost. The second one proposes a splitting method applied to the two-speed relaxation system for the Euler system, whose acoustic part involves only constant coefficients. This leads to implicit steps with controlled conditioning.Finally, the fourth chapter regards the development of models for collective dynamics of cells. We present three different models with specific interaction rules: layer interactions, back-to-back anti-alignment and interactions with surrounding elastic cables. We study their large-scale dynamics either theoretically, by deriving the associated macroscopic models, or using numerical simulations. Comparisons with experimental results have also been performed.