Research

Numerical scheme derivations to simulate self-organized collective motions

My thesis focuses on the study and numerical simulation of self-organized collective motions. We particularly focus on the Vicsek microscopic model (individual scale) and to its associated macroscopic model (population density scale). They describe, for instance, the motion of insects or birds and replicate the emergence of a self-organized collective motion.

This thesis firstly aims at understanding better the macroscopic derivative of the Vicsek model, called the Self-Organized Hydrodynamics (SOH) model and to develop an efficient and robust numerical scheme. The main difficulty resides in the fact that the model is hyperbolic and non-conservative. Indeed, the shock wave solutions of the model ar not well-defined. In addition, the numerical approximations of these solutions do not converge to the exact solution. Hence, first of all, shock wave solutions were defined as imits of traveling wave solutions of the viscous SOH model. This definition allowed us to derive generalizec Rankine-Hugoniot conditions for the SOH model. Secondly, a Godunov-type scheme was derived for the SOH model, to which a numerical viscous correction was added in order to capture properly shocks. This work is the subject of a first paper (HAL).

This research project then seeks to get more realistic simulations. On the one hand, a micro-macro decomposition method will allow us to conserve kinetic aspects in the model as well as having a computation cost cheaper than a particle method. Indeed, the SOH model takes into account only motions where all individuals are part of the equilibrium. A micro-macro decomposition will allow us to keep a kinetic contribution of the deviation to the equilibrium. On the other hand, we aim at enriching the macroscopic model in order to take into account fear or appetence phenomena inside the population.

My PhD supervisors are Christophe Berthon and Anaïs Crestetto.

Keywords:

Preprints

M. Compain, C. Berthon, A. Crestetto. A micro-macro numerical scheme for a self-organized hydrodynamics model enriched by a kinetic contribution. (2026) HAL

Abstract

The present work investigates the continuous individual based Couzin-Vicsek model, the associated kinetic model and its macroscopic derivation, called the Self-Organized Hydrodynamics (SOH) model. They describe self-organized motions, which emerge in nature as insect swarms or bird flocks for instance. In particular, the SOH model takes into account only motions where all individuals are part of the equilibrium. Here, according to a micro-macro decomposition, we enrich the model with a kinetic contribution of the deviation from equilibrium. First of all, a micro-macro model is derived. We then design an asymptotic preserving numerical scheme for this model, where the equilibrium variables are discretized thanks to a finite volume method, while the kinetic deviation is approximated with a particle method. Finally, some test cases are computed, and the micro-macro scheme is validated in different asymptotic regimes.

M. Compain, C. Berthon, A. Crestetto. A shock-capturing numerical scheme for a non-conservative self-organized hydrodynamics model. (2026) HAL

Abstract

This work focuses on the Self-Organized Hydrodynamics (SOH) model, which is the macroscopic limit of the well-known individual based Vicsek model. The SOH model is a hyperbolic non-conservative PDE system with a geometric constraint, which causes issues for both its theoretical and numerical resolutions. In this work, we first reformulate the model without its constraint for some well-prepared initial conditions. We then focus on shock wave solutions of the SOH model and we define generalized Rankine-Hugoniot conditions. Moreover, a Godunov-type scheme is designed and a viscous correction is added in order to numerically recover shock wave solutions. Finally, some exact solutions to the Riemann problem are computed thanks to the generalized Rankine-Hugoniot conditions. Thus the shock-capturing scheme is compared to an usual splitting method and the exact solutions on some test cases. These simulations confirm the relevance of the viscous Godunov-type scheme to simulate shock waves of the SOH model.

Riemann solver for the SOH model

You can get exact solutions for the Riemann problem associated to the SOH model here: