Francesco Salvarani

Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method

We study the two-scale asymptotics for a charged beam under the action of a rapidly oscillating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution.

1D nonlinear Fokker–Planck equations for fermions and bosons

Abstract We obtain equilibration rates for nonlinear Fokker–Planck equations modelling the relaxation of fermion and boson gases. We show how the entropy method applies for quantifying explicitly the exponential decay towards Fermi–Dirac and Bose–Einstein distributions in the one-dimensional case.

The nonlinear diffusion limit for generalized Carleman models: the initial-boundary value problem

Consider the initial-boundary value problem for the 2-speed Carleman model of the Boltzmann equation of the kinetic theory of gases, (see Carleman 1957 Problemes Mathematiques Dans la Theorie Cinetique des Gaz (Uppsala: Almqvist-Wiksells)), set in some bounded interval with boundary conditions prescribing the density of particles entering the interval. Under the usual parabolic scaling, a nonlinear diffusion limit is established for this problem. In fact, the techniques presented here allow treatment generalizations of the Carleman system where the collision frequency is proportional to the αth power of the macroscopic density, with α [−1, 1].

The diffusive limit for Carleman-type kinetic models

We study the limiting behaviour of the Cauchy problem for a class of Carleman-like models in the diffusive scaling with data in the spaces Lp, 1 ≤ p ≤ ∞. We show that, in the limit, the solution of such models converges towards the solution of a nonlinear diffusion equation with initial values determined by the data of the hyperbolic system. When the data belong to L1, a condition of conservation of mass is needed to uniquely identify the solution in some cases, whereas the solution may disappear in the limit in other cases.

Modelling opinion formation by means of kinetic equations

In this chapter, we review some mechanisms of opinion dynamics that can be modelled by kinetic equations. Beside the sociological phenomenon of compromise, naturally linked to collisional operators of Boltzmann kind, many other aspects, already mentioned in the sociophysical literature or no, can enter in this framework. While describing some contributions appeared in the literature, we enlighten some mathematical tools of kinetic theory that can be useful in the context of sociophysics.New opinions are always suspected, and usually opposed, without any other reason but because they are not already common.John Locke, An Essay Concerning Human Understanding

On the Optimal Choice of Coefficients in a Truncated Wild Sum and Approximate Solutions for the Kac Equation

We study an approximate solution of the Boltzmann problem for Kacs caricature of a Maxwellian gas by using a truncated and modified expansion of Wild type. We choose the coefficients in the Wild sum approximation using a criterion based on exactly reproducing the behavior of the leading modes.

GPU-accelerated numerical simulations of the Knudsen gas on time-dependent domains

Abstract We consider the long-time behaviour of a free-molecular gas in a time-dependent vessel with an absorbing boundary, in any space dimension. We first show, at the theoretical level, that the convergence towards equilibrium heavily depends on the initial data and on the time evolution law of the vessel. Subsequently, we describe a numerical strategy to simulate the problem, based on a particle method implemented on general-purpose graphics processing units (GPGPU). We observe that the parallelisation procedure on GPGPU allows for a marked improvement of the performances when compared with the standard approach on CPU.

Large-time asymptotics for nonlinear diffusions: the initial-boundary value problem

In this paper we investigate the large-time behavior of solutions to the first initial-boundary value problem for the nonlinear diffusion ut=(um)xx,m>0. In particular, we prove exponential decay of u(x,t) towards its own steady state in L1-norm for long times and we give an explicit upper bound for the rate of decay. The result is based on a new application of entropy estimates, and on detailed lower bounds for the entropy production in this situation.

APPROXIMATED SOLUTIONS OF PHOTON TRANSPORT IN A TIME DEPENDENT REGION

The paper is devoted to the study of a transport linear integro-differential equation in a time-dependent domain with slab geometry. After proving existence and uniqueness results, we show that our problem admits an approximated (quasi-static) solution, and evaluate the error between the exact solution and its approximation, which turns out to be good under suitable assumptions. This permits, in all practical calculations, the use of an equation, which is simpler because the time variable may be considered as parameter. At the end of the paper some numerical simulations are performed.

Exponential relaxation to self-similarity for the superquadratic fragmentation equation

We consider the self-similar fragmentation equation with a superquadratic fragmentation rate and provide a quantitative estimate of the spectral gap.