Discontinuous-galerkin methods for a kinetic model of self-organized dynamics

TL;DR

This paper develops and analyzes a discontinuous Galerkin numerical method for solving kinetic models of self-organized particle systems with alignment and attraction-repulsion interactions, validated through various numerical experiments.

Contribution

It introduces a novel DG-spectral scheme for kinetic models with non-local interactions, including stability and consistency analysis.

Findings

The method accurately captures self-organized dynamics.

Numerical experiments confirm stability and consistency.

The approach effectively handles non-local drift velocities.

Abstract

This paper deals with the numerical resolution of kinetic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. We focus on the kinetic model considered in [18, 17] where alignment is taken into account in addition of an attraction-repulsion interaction potential. We apply a discontinuous Galerkin method for the free transport and non-local drift velocity together with a spectral method for the velocity variable. Then, we analyse consistency and stability of the semi-discrete scheme. We propose several numerical experiments which provide a solid validation of the method and its underlying concepts.