Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit

Giacomo Dimarco, Lorenzo Pareschi, Giovanni Samaey (30/7/2017 preprint arXiv:1707.09672)

We develop a new Monte Carlo method that solves hyperbolic transport equations with stiff terms, characterized by a (small) scaling parameter. In particular, we focus on systems which lead to a reduced problem of parabolic type in the limit when the scaling parameter tends to zero. Classical Monte Carlo methods suffer of severe time step limitations in these situations, due to the fact that the characteristic speeds go to infinity in the diffusion limit. This makes the problem a real challenge, since the scaling parameter may differ by several orders of magnitude in the domain.

Uncertainty quantification for kinetic models in socio-economic and life sciences

Giacomo Dimarco, Lorenzo Pareschi, Mattia Zanella
(26/6/2017 preprint arXiv:1706.17500) to appear in "Uncertainty quantification for kinetic and hyperbolic equations" SEMA-SIMAI Springer Series.

Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic.

Efficient Stochastic Asymptotic-Preserving IMEX Methods for Transport Equations with Diffusive Scalings and Random Inputs

Shi Jin, Hanqing Lu, Lorenzo Pareschi
(14/3/2017 preprint arXiv:1703.03841)

For linear transport and radiative heat transfer equations with random inputs, we develop new generalized polynomial chaos based Asymptotic-Preserving stochastic Galerkin schemes that allow efficient computation for the problems that contain both uncertainties and multiple scales.

Structure preserving schemes for nonlinear Fokker-Planck equations and applications

Lorenzo Pareschi, Mattia Zanella
(1/2/2017, preprint arXiv:1702.00088)

In this paper we focus on the construction of numerical schemes for nonlinear Fokker-Planck equations that preserve the structural properties, like non negativity of the solution, entropy dissipation and large time behavior. The methods here developed are second order accurate, they do not require any restriction on the mesh size and are capable to capture the asymptotic steady states with arbitrary accuracy.