Laurent Navoret

Pierre Degond, Jiale Hua, Laurent Navoret

In this paper, we study the numerical simulations for Euler system with maximal density constraint. This model is developed in [1, 3] with the constraint introduced into the system by a singular pressure law, which causes the transition of different asymptotic dynamics between different regions. To overcome these difficulties, we adapt and implement two asymptotic preserving (AP) schemes originally designed for low Mach number limit [2,4] to our model. These schemes work for the different dynamics and capture the transitions well. Several numerical tests both in one dimensional and two dimensional cases are carried out for our schemes.

Pierre Degond, Piotr Minakowski, Laurent Navoret et al.

We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure. This model describes for instance the macroscopic motion of a crowd with individual congestion preferences. We propose an asymptotic preserving (AP) scheme based on a conservative formulation of the system in terms of density, momentum and density fraction. A second order accuracy version of the scheme is also presented. We validate the scheme on one-dimensional test-cases and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics.

David Coulette, Emmanuel Franck, Philippe Helluy et al.

We present a high order scheme for approximating kinetic equations with stiff relaxation. The objective is to provide efficient methods for solving the underlying system of conservation laws. The construction is based on several ingredients: (i) a high order implicit upwind Discontinuous Galerkin approximation of the kinetic equations with easy-to-solve triangular linear systems; (ii) a second order asymptotic-preserving time integration based on symmetry arguments; (iii) a palindromic composition of the second order method for achieving higher orders in time. The method is then tested at orders 2, 4 and 6. It is asymptotic-preserving with respect to the stiff relaxation and accepts high CFL numbers.

Mehdi Badsi, Michel Mehrenberger, Laurent Navoret

We are interested in developing a numerical method for capturing stationary sheaths, that a plasma forms in contact with a metallic wall. This work is based on a bi-species (ion/electron) Vlasov-Amp{è}re model proposed in [3]. The main question addressed in this work is to know if classical numerical schemes can preserve stationary solutions with boundary conditions, since these solutions are not a priori conserved at the discrete level. In the context of high-order semi-Lagrangian method, due to their large stencil, interpolation near the boundary of the domain requires also a specific treatment.

Laurent Navoret, Roxana Sublet, Marcela Szopos

The goal of the present work is to propose an agent-based model that originally combines classical Vicsek-like polarity alignments and contact forces, as implemented in the framework developed by Maury and Venel in [Maury, Venel, 2011]. The description additionally incorporates velocity feedback on polarity and soft attraction-repulsion interactions. After carefully studying the well posedness of the model, we introduce a suitable discretization and perform an extensive range of numerical experiments to assess the impact of different modeling ingredients. The dynamical system is capable of recovering the order-disorder phase transition of the flock, as well as the jamming effect in high density regimes. As such, the developed framework can be seen as a promising theoretical tool that could contribute to improving the understanding of complex collective cell dynamics and emerging tissue flows.

Clémentine Courtès, Emmanuel Franck, Michael Kraus et al.

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.