Uncertainty Quantification in Control Problems for Flocking Models

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella 
(5/3/2015 Math. Probl. Eng.  (2015), Art. ID 850124, 14 pp.,  arXiv:1503.00548)

In this paper the optimal control of flocking models with random inputs is investigated from a numerical point of view. The effect of uncertainty in the interaction parameters is studied for a Cucker-Smale type model using a generalized polynomial chaos (gPC) approach. Numerical evidence of threshold effects in the alignment dynamic due to the random parameters is given.

Fast solvers for the quantum Boltzmann equation and the Bose-Einstein condensation

Bose-Einstein condensate

The interest in the quantum framework of the Boltzmann equation has in­creased dramatically in the recent years. Although the quantum Boltzmann equation (QBE) for a single specie of particles is valid for a gas of fermions as well as for a gas of bosons, blow up of the solution in finite time may occur only in the latter case. In particular this equation has been successfully used for computing non­equilibrium situations where Bose­Einstein condensate occurs.

Hybrid multiscale methods for hyperbolic relaxation systems

2D Burgers limit

Hyperbolic systems with relaxation arise in a wide variety of physical problems, ranging from linear and nonlinear waves, kinetic theory, to multiphase and phase transition modeling. We report here some numerical results of one-dimensional and two-dimensional computations using hybrid stochastic-deterministic methods of problems involving different scales.

Time Relaxed Monte Carlo (TRMC) simulation of the Boltzmann equation

Shuttle reentry

Computation of the aerothermodynamics of hypersonic re-entry vehicles along their entire trajectory involves continuum conditions at low altitudes and rarefied, or non-equilibrium, conditions at high altitudes. Accurate 2D and 3D DSMC simulations can require prohibitively high numbers of simulated particles, especially in regions where the Knudsen number is very small.

Numerical solution of the Fokker-Planck-Landau equation by fast spectral methods

Kinetic density

Nowadays, numerical simulations of plasmas are receiving a great deal of attention both in research and in industry thanks to the numerous applications directly connected to these phenomena. In addition, there exist many practical situations in which the so-called Coulomb collisions are fundamental for correctly describing the plasma dynamics as for instance in magnetic fusion devices (like tokamak devices). The Fokker-Planck-Landau (FPL) equation is used to describe the binary collisions between charged particles in plasma physics.

Relaxation approximations and numerical solution of degenerate diffusion equations

Hele-Shaw cell

Nonlinear diffusion equations come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry, thin films and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the development of numerical schemes more involved and challenging.

Numerical approximation of kinetic equations for granular gases

Clusters formation

A granular gas is a conglomeration of discrete solid, particles characterized by a loss of energy whenever the particles interact. Unlike conventional gases granular materials will tend to cluster and clump due to the dissipative nature of the collisions between grains which may be expressed by a decay of its granular temperature. Applications include the handling and storage of cereals, granular chemicals, sand, coal, pharmaceuticals, and more.