Narrow Results

We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first-order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly nonlocal interaction, we derive diffusive corrections to the first-order system which lead to the combination of a heat flow of the harmonic map and Landau–Lifschitz–Gilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological Landau–Lifschitz–Gilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics.

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 [P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci.18 (2008) 1193–1215; P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal.20 (2013) 89–114] where alignment is taken into account in addition to 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 analyze consistency and stability of the semi-discrete scheme. We propose several numerical experiments which provide a solid validation of the method and illustrate its underlying concepts.