J. A. Carrillo

J. A. Carrillo, L. Pareschi, M. Zanella

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean--field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.

J. A. Carrillo, Y. Huang, F. S. Patacchini et al.

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level, and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

J. A. Carrillo, A. Klar, A. Roth

We show that double mills are more stable than single mills under stochastic perturbations in swarming dynamic models with basic attraction-repulsion mechanisms. In order to analyse accurately this fact, we will present a numerical technique for solving kinetic mean field equations for swarming dynamics. Numerical solutions of these equations for different sets of parameters will be presented and compared to microscopic and macroscopic results. As a consequence, we numerically observe a phase transition diagram in term of the stochastic noise going from single to double mill for small stochasticity fading gradually to disordered states when the noise strength gets larger. This bifurcation diagram at the inhomogeneous kinetic level is shown by carefully computing the distribution function in velocity space.

V. Bonnaillie-Noël, J. A. Carrillo, T. Goudon et al.

In this paper we study the diffusion approximation of a swarming model given by a system of interacting Langevin equations with nonlinear friction. The diffusion approximation requires the calculation of the drift and diffusion coefficients that are given as averages of solutions to appropriate Poisson equations. We present a new numerical method for computing these coefficients that is based on the calculation of the eigenvalues and eigenfunctions of a Schrödinger operator. These theoretical results are supported by numerical simulations showcasing the efficiency of the method.