Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation

Giacomo Albi, Giacomo Dimarco, Lorenzo Pareschi
(8/4/2019 preprint arXiv:1904.03865)

We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system.

Hydrodynamic models of preference formation in multi-agent societies

Lorenzo Pareschi, Giuseppe Toscani, Andrea Tosin, Mattia Zanella
(24/12/2018, preprint arXiv:1901.00486)

In this paper, we discuss the passage to hydrodynamic equations for kinetic models of opinion formation. The considered kinetic models feature an opinion density depending on an additional microscopic variable, identified with the personal preference. This variable describes an opinion-driven polarisation process, leading finally to a choice among some possible options, as it happens e.g. in referendums or elections.

Multi-scale variance reduction methods based on multiple control variates for kinetic equations with uncertainties

Giacomo Dimarco, Lorenzo Pareschi
(12/12/2018,  preprint arXiv:1812.05485)

The development of efficient numerical methods for kinetic equations with stochastic parameters is a challenge due to the high dimensionality of the problem. Recently we introduced a multiscale control variate strategy which is capable to accelerate considerably the slow convergence of standard Monte Carlo methods for uncertainty quantification. Here we generalize this class of methods to the case of multiple control variates.

Multi-scale control variate methods for uncertainty quantification in kinetic equations

Giacomo Dimarco, Lorenzo Pareschi
(25/10/2018. J. Comp. Phys to appear arXiv:1810.10844)

Kinetic equations play a major rule in modeling large systems of interacting particles. Uncertainties may be due to various reasons, like lack of knowledge on the microscopic interaction details or incomplete informations at the boundaries. These uncertainties, however, contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge.